Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

S S symmetry

Article Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

Mike Behrisch 1,*,† and Edith Vargas-García 2,†

1 Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Wiedner Hauptstr. 8–10, 1040 Vienna, Austria 2 Departamento Académico de Matemáticas, ITAM, Río Hondo No. 1, Col. Tizapán San Angel, Del. Álvaro Obregón, Ciudad de México C.P. 01080, Mexico; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work.

Abstract: As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

Keywords: centralising monoid; centraliser clone; commutation; endoprimal monoid; endomorphism monoid; maximal centralising monoid

MSC: 08A35; 08A02; (08A40, 08A60)

Citation: Behrisch, M.; Vargas-García, E. Centralising Monoids with Low-Arity Witnesses 1. Introduction on a Four-Element Set. Symmetry 2021, 13, 1471. https://doi.org/ There are various notions of symmetry in the sciences and in mathematics. Algebraic 10.3390/sym13081471 structures are usually considered symmetric if they have a lot of automorphisms or, more generally, endomorphisms. For a universal algebra (a structure without relations) the Academic Editor: Ivan Chajda automorphism group/endomorphism monoid consists of all those permutations/self- maps of the carrier set that commute with all fundamental operations of the algebra. Received: 23 July 2021 Studying commuting operations in more generality leads to the notion of the centraliser Accepted: 5 August 2021 clone of an algebra, or simply of a set of operations if the carrier set is understood from Published: 11 August 2021 the context. The unary part of the centraliser (clone) is then exactly the endomorphism monoid of the algebra, and the fundamental operations of the algebra are said to witness Publisher’s Note: MDPI stays neutral this monoid. with regard to jurisdictional claims in This article is concerned with algebras on a specific carrier set A that are ‘maximally published maps and institutional affil- symmetric’ in the sense that their endomorphism monoid is a co-atom in the lattice of all iations. possible endomorphism monoids of algebras on A. As the monoids in this lattice are the unary parts of (all) centraliser clones on A, they are called centralising monoids, and the co-atoms of this lattice are also referred to as maximal centralising monoids. The study of centralisers in algebra goes back to Cohn [1] (Chapter III.3), and notably Copyright: © 2021 by the authors. to Kuznecov [2] who first established logical methods for their investigation, exploiting Licensee MDPI, Basel, Switzerland. closure under operations whose graphs are primitive positively definable from given This article is an open access article operations. Kuznecov allegedly also discovered all 25 centraliser clones among Post’s distributed under the terms and lattice, a fact later re-proved by Hermann [3]. Danil’ˇcenko[4–7] continued the work of conditions of the Creative Commons Kuznecov by determining all 2986 centraliser clones on sets of three elements [8,9]. On Attribution (CC BY) license (https:// carrier sets of size four and beyond, currently no good overview of the lattice of centralisers creativecommons.org/licenses/by/ exists. Between 1974 and 1976 Harnau [10–15] worked on centralisers of unary operations, 4.0/).

Symmetry 2021, 13, 1471. https://doi.org/10.3390/sym13081471 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1471 2 of 40

which are dual to centralising monoids in terms of the Galois connection induced by commutation of finitary operations (see Section2). Centralising monoids of single unary operations, i.e., monounary algebras, were investigated in [16–18], showing, for example, which centralising monoids of this type are equal to the centralising monoid they describe as a witness [16] (Theorem 4.1, p. 8, Theorem 5.1, p. 10), and which of them have a unique unary operation as their witness [18] (Theorems 3.1 and 3.3, p. 4659 et seq.). Research on centralising monoids was further pushed forward in a series of papers [19–25] by Machida and Rosenberg, linking in particular maximal centralising monoids to the five types of functions appearing in Rosenberg’s Classification Theorem [26] for minimal clones. As a consequence, all 192 centralising monoids on three-element sets were determined in [21,23], and all 10 maximal centralising monoids among them were identified. This research is part of the project, begun in [27], to take the results of [23] to the next level, that is, to determine all maximal centralising monoids on four-element carrier sets. According to Proposition3 it is necessary for this to find all centralising monoids on {0, 1, 2, 3} induced by single functions of each of the five types of Rosenberg’s Theo- rem and to determine the ones among them which are maximal proper transformation monoids under set inclusion. As an initial step centralising groups of majority operations and semiprojections were studied in [28]. Extending the methods of [23], the case of majority operations on {0, 1, 2, 3} was already completed in [27], and further investigated under a different aspect in [29]. In this article we tackle the cases of unary operations (permutations of prime order and non-identical retractive operations), binary idempotent operations and ternary minority operations arising as Mal’cev operations corresponding to Boolean groups. We determine all centralising monoids on 4 = {0, 1, 2, 3} witnessed by binary idempotent operations, and we classify both the witnessing operations and the monoids up to conjugacy by inner automorphisms from Sym 4. We also exhibit the maximal monoids among them, in general, and again, up to conjugacy. With respect to unary operations, we establish that on every carrier set of size at least four, every single transposition or every product of disjoint transpositions without fixed points and every non-identical retraction witnesses a centralising monoid which is a co-atom in the lattice of those centralising monoids witnessed by sets of unary operations, that is, in the lattice of endomorphism monoids of unary algebras. However, we also show that, given at least four elements, no transposition and no permutation with a single fixed point, as well as no retraction fixing all but one element can ever witness a maximal centralising monoid. For {0, 1, 2, 3} we improve this by showing that no permutation at all will witness a maximal centralising monoid. Finally, based on results of Länger and Pöschel [30] about strongly constantively rigid relational systems, we give a new proof of the known fact [21] (Theorem 5.1) that the centralising monoid of a constant on an at least three-element set is indeed maximal. We use the same technique to prove that—with the exception of the two-element set—the centralising monoid of a Mal’cev operation of a Boolean group is always maximal.

2. Preliminaries We start by introducing basic notation with respect to sets, functions and relations, followed by fundamental facts regarding their Galois theory based on preservation and commutation. We also give a brief overview of formal concept analysis in order to be able to switch between different Galois connections. In the second part of this section we present background theory on clones, in particular centraliser clones, and centralising monoids that we shall use extensively to derive our main results.

2.1. Notation and Basic Concepts We write N = {0, 1, 2 . . . } for the set of all natural numbers (ﬁnite ordinals) and N+ for the positive ones. It will be convenient for us to understand every n ∈ N as the set of its ordinal predecessors n = {0, . . . , n − 1} as in the von Neumann model of natural Symmetry 2021, 13, 1471 3 of 40

numbers. The cardinality (size) of a set A is denoted by |A|; if |A| < ℵ0, then we often pick a canonical representative for it, e.g., n = {0, . . . , n − 1} where n = |A|. Given sets A, B, C and functions f : A −→ B and g : B −→ C, we denote their composition by g ◦ f : A −→ C and mean the function mapping each element a ∈ A to (g ◦ f )(a) := g( f (a)) ∈ C. The set of all functions from A to B is symbolised as BA. If f ∈ BA and U ⊆ A, V ⊆ B, we write f [U] for the image { f (u) | u ∈ U} and f −1[V] for the preimage { x ∈ A | f (x) ∈ V}. The full image of f is also denoted as im( f ) := f [A]. The restriction of f to an arbitrary subset U ⊆ A is the function f |U : U −→ B mapping x ∈ U to f |U(x) := f (x). For n ∈ N we understand tuples x ∈ An as maps from n = {0, . . . , n − 1} to A that are simply written as x = (x0, ... , xn−1). Sometimes we allow ourselves to deviate from this standard notation if some other indexing like x = (a, b, c) or x = (x1, ... , xn) seems more n convenient. As tuples x ∈ A are maps, we have in particular im(x) = {x0,..., xn−1} and A I n we can compose with f ∈ B or α ∈ n , giving f ◦ x = ( f (x0), ... , f (xn−1)) ∈ B and I re-indexed tuples x ◦ α = xα(i) ∈ A . i∈I (n) An For a set A and n ∈ N, every f ∈ OA := A is said to be an n-ary operation on A. O = S O(n) We collect all finitary (non-nullary) operations on A in the set A n∈N+ A . For (n) (n) (n) (n) F ⊆ OA and n ∈ N we write F for F ∩ OA ; in particular OA = (OA) for n > 0. (n) (n) For i ∈ n ∈ N the n-ary projection onto the coordinate i is the operation ei ∈ OA given by (n) n (1) e (x) := xi for x ∈ A . We write shortly idA := e for the identity operation on A, and i n o 0 J = (n) ∈ ∈ ∈ we use A ei i n N for the set of all projections. For a A the n-ary constant (n) (n) (n) n with value a is the map ca ∈ O satisfying ca (x) = a for all x ∈ A . We collect all A n o (n) constant operations on A in the set CA = ca a ∈ A ∧ n ∈ N+ . We also need some more specific operations given by identities. An operation f ∈ OA is idempotent, if it satisfies (3) f (x, ... , x) = x for all x ∈ A. A ternary operation f ∈ OA is called a majority operation, if f (x, x, y) = f (x, y, x) = f (y, x, x) = x for all x, y ∈ A; it is called a minority operation if f (x, x, y) = f (x, y, x) = f (y, x, x) = y for all x, y ∈ A; finally, it is a Mal’cev operation if f (x, x, y) = f (y, x, x) = y for all x, y ∈ A. Among unary operations we need permutations (1) (1) f ∈ Sym A, i.e., bijective self-maps f ∈ OA , and retractive operations f ∈ OA satisfying f ◦ f = f (sometimes also called idempotent since they are idempotent elements in the D (1) E semigroup OA ; ◦ ). (m) m n For all m, n ∈ N and g1, ... , gn ∈ OA , we can form (g1, ... , gn) : A −→ A , called m n the tupling of g1, ... , gn, sending x ∈ A to (g1(x), ... , gn(x)) ∈ A . In this way, we can (m) (n) compose finitary operations g1, ... , gn ∈ OA with f ∈ OA as f ◦ (g1, ... , gn).A (concrete) clone of non-nullary operations on A is any set F ⊆ OA that is closed under this form of composition and satisfies JA ⊆ F. For m ∈ N an m-ary relation on a set A is any subset $ ⊆ Am of m-tuples. Any (m) m-ary operation f ∈ OA can be understood as an (m + 1)-ary relation via its graph • f := { (x1,..., xm, g(x1,..., xm)) | x1,..., xm ∈ A}. We extend this notation element- • • wise to sets of operations by putting F := { f | f ∈ F} for F ⊆ OA. A binary relation $ ⊆ A2 is just a set of pairs; we define its inverse (sometimes also called dual) to be $−1 := { (y, x) | (x, y) ∈ $}. A binary $ ⊆ A2 is symmetric if $−1 ⊆ $, and reflexive if (x, x) ∈ $ for all x ∈ A. If $ is reflexive, symmetric and transitive, that is (x, y), (y, z) ∈ $ implies (x, z) ∈ $ for all x, y, z ∈ A, then $ belongs to Eq(A), the set of all equivalence m relations on A. By RA = { $ ⊆ A | m ∈ N+} we denote the set of all finitary relations on A. While functions can be composed, new relations can be constructed from given ones using logical expressions. A primitive positive formula ϕ (in prenex normal form) over a given relational signature consists of a prefix of finitely many existentially quantified variables (possibly none) followed by a finite non-empty conjunction of atomic predicates that correspond to the given signature (relations) and have been substituted by some Symmetry 2021, 13, 1471 4 of 40

tuple of variables. If the relation symbols are associated with concrete relations, the set of all satisfying value assignments to (a superset of) the free variables of ϕ determines a finitary relation on A, which is called primitive positively definable from the given relations. For a finite carrier set A, a set Q ⊆ RA is said to be a relational clone on A if it contains the diagonal ∆A := { (x, x) | x ∈ A} and is closed under all relations that are primitive positively definable from members of Q (for infinite carrier sets stronger closure properties are required). As this is an implicational definition, all relational clones on A form a closure system. The least relational clone on A containing a particular set R ⊆ RA will be denoted by [R] and is computed by adding to R all relations that are primitive positively RA definable from R ∪ {∆A}. Clones of operations and relational clones are connected in the following way: for m, n ∈ N and f : An −→ A, $ ⊆ Am we say that f preserves $ if and only if $ is a subuniverse m n of the algebra hA; f i . This means that for every (r0, ... , rn−1) ∈ $ the m-tuple obtained by the composition f ◦ (r0, ... , rn−1) of f with the tupling of r0, ... , rn−1 stays inside $, that is, for every (m × n)-matrix R ∈ Am×n the columns of which all belong to $, the m-tuple obtained by applying f row-wise to R remains in $. We then call $ an invariant of f or f a polymorphism of $. The preservation relation induces a Galois connection between OA and RA, giving rise to the derivation operators Q 7→ PolA Q and F 7→ InvA F, collecting all polymorphisms in OA of every relation $ ∈ Q ⊆ RA and all invariant relations $ ∈ RA for all given operations f ∈ F ⊆ OA, respectively. Moreover, we declare for n ∈ N and (n) (n) (n) (n) F ⊆ OA, Q ⊆ RA that PolA F := (PolA F) and InvA Q := (InvA Q) . Now for every Q ⊆ RA the set PolA Q forms a clone on A, and for every F ⊆ OA the invariants InvA F are a relational clone containing ∅ (since we omit nullary operations in our clones). Therefore, Inv Pol Q ⊇ [Q ∪ {∅}] and Pol Inv F contains the clone generated by F. On finite A A RA A A carrier sets A, these inclusions are equalities [31,32]; for infinite A local interpolation operators (and a strengthened definition of relational clone) need to be added to close the gap [33,34]. Besides the Galois connection induced by preservation, we shall encounter several other Galois connections that differ by the inducing relation or by restrictions of the domains they link. Switching between them and the associated closure systems can be expressed well in the framework of formal concept analysis [35,36], providing terminology, notation and theory for the manipulation of Galois connections on a general level. The basic object of formal concept analysis is that of a formal context K = (G, M, I) where I ⊆ G × M is any binary relation between sets G (commonly called objects) and M (usually called attributes). A context K induces two derivation operators of a Galois connection between G and M in the natural way, and specifies exactly between which sets this Galois connection is to be understood. The Galois closed sets on the side of objects are called extents and the ones on the side of attributes are referred to as intents. When we keep the relation I, but pass on to a subset H ⊆ G of the objects or N ⊆ M of the attributes (or both), we form 0 a subcontext K = (H, N, I ∩ (H × N)). Though technically incorrect, it is customary to omit the intersection ∩ (H × N) when specifying a subcontext, as the restriction becomes clear from stating the sets of objects and attributes. How the lattices of Galois closed sets 0 (extents and intents) of K and K are related is discussed in Chapter 3 of [35] (p. 97 et seqq.). We will mostly aim for context manipulations where the closure system of intents (and thus the lattice structure) does not change, such as object clarification and object reduction. We shall explain more details at the appropriate place in the text and give pointers to the literature there. Centrally for this paper will be the Galois connection of commutation given by the (n) context Kc = (OA, OA, ⊥). In this respect we say that an n-ary operation f ∈ OA (m) T commutes with an m-ary operation g ∈ OA , if g( f (X)) = f g X holds for any matrix m×n X ∈ A , where f (X) := ( f (X(i, ·)))i∈m denotes the tuple obtained by applying f row- wise to the matrix X, and similarly for g and the transposed matrix XT. This commutation condition will be denoted by f ⊥ g; it is easy to see that it is a symmetric property, i.e., we have f ⊥ g if and only if g ⊥ f . Therefore, the two Galois derivatives induced by Kc Symmetry 2021, 13, 1471 5 of 40

∗ coincide: they map F ⊆ OA to its centraliser F = { g ∈ OA | ∀ f ∈ F : g ⊥ f }, and the associated closure operator sends F to F∗∗, the bicentraliser of F. A routine verification ∗ • • shows that g ∈ F ⇐⇒ g ∈ PolA F ⇐⇒ g ∈ InvA F holds for F ∪ {g} ⊆ OA, so ∗ • F = PolA F , i.e., commutation can be rephrased in terms of preservation of graphs of operations. Hence, up to isomorphism, Kc is the subcontext of finitary operations and • relations with preservation, where RA is restricted to [OA] . As mentioned in the introduction, when studying centralising monoids we only look at the unary parts F∗(1) of centraliser clones, that is to say, we are dealing with the (1) (1) (1) subcontext K := OA, OA , ⊥ ∩ OA × OA , briefly OA, OA , ⊥ , of Kc. The intents ∗(1) of K, i.e., Galois closed sets M = F for some F ⊆ OA, are exactly all centralising monoids on A. Moreover, every set F ⊆ OA describing M in this way, is a witness of M, and we can equivalently express that M is witnessed by F via saying that M = End hA; Fi.A (1) centralising monoid M ⊆ OA is maximal if it is a co-atom in the intent lattice of K, i.e., a proper centralising monoid which is maximal under set inclusion.

2.2. Fundamental Results on Clones, Centralisers and Centralising Monoids The following simple observation exhibits necessary conditions that can be used to describe functions in a particular centraliser clone (or its unary part). The utility of said conditions was already observed by Harnau, cf. Lemma 2.6 and Satz 2.15 of [10], in the (1) context of centralisers of single unary operations f ∈ OA .

Lemma 1. For every set F ⊆ O we have F∗ = Pol [F•] . In particular, we have A A RA

∗ F ⊆ PolA{ ker( f ) | f ∈ F}, ∗ F ⊆ PolA{ im( f ) | f ∈ F}, ∗ F ⊆ PolA{ ﬁx( f ) | f ∈ F},

(n) where for n ∈ N and f ∈ OA n o 2n ker( f ) := (x1,..., xn, y1,..., yn) ∈ A f (x1,..., xn) = f (y1,..., yn) , im( f ) := f [An], ﬁx( f ) := { x ∈ A | f (x,..., x) = x}.

Proof. We have F∗ = Pol F• ⊇ Pol [F•] ⊇ Pol Inv Pol F• = Pol F•, following A A RA A A A A directly from the definition of commutation. Clearly, fix( f ) is primitive positively definable from the graph of f , im( f ) = { y ∈ A | ∃x1,..., xn ∈ A : f (x1,..., xn) = y} is too, and ker( f ) = (x, y) ∈ A2n ∃z ∈ A : f (x) = z ∧ f (y) = z as well. Second, we recollect the following helpful characterisation of centralising monoids.

(1) Lemma 2 (cf. [21] (Lemma 2.2)). For a set M ⊆ OA the following facts are equivalent: ∗(1) (a) M is a centralising monoid, i.e., there is F ⊆ OA such that M = F . (b) M = M∗∗(1), i.e., M is witnessed by M∗.

∗(1) ∗∗(1) ∗(1) ∗ ∗(1) ∗(1) Proof. If M = F for some F ⊆ OA, then M = ((F ) ) = F = M by virtue of the Galois derivatives ∗ and ∗(1). Conversely, if M = M∗∗(1), then M is a centralising monoid witnessed by F = M∗.

Lemma2 shows that every centralising monoid M = F∗(1) has a largest possible witness, as generally M ⊆ F∗, so F ⊆ M∗. We now turn to maximal centralising monoids. (1) Being co-atoms in the lattice of intents of K = OA, OA , ⊥ , they correspond via that Galois connection of commutation to an atom F in the lattice of extents of K. Such an Symmetry 2021, 13, 1471 6 of 40

(1)∗ atom must be generated (as an extent) by a single non-trivial object f ∈ OA \OA , i.e., F = { f }∗(1)∗. For the co-atom M = F∗(1) this means M = F∗(1) = { f }∗(1)∗∗(1) = { f }∗(1) by virtue of the Galois connection given by K. So M is singly witnessed by a non-trivial operation f ; however, even more is known about these witnesses: they can always be chosen as a generator of a minimal clone of minimum arity, a so-called minimal function.

Proposition 3 (Theorem 3.2 in [21], Proposition 2.2 in [27]). For any maximal centralising (1) ∗ (1)∗ monoid M ⊆ OA on a ﬁnite set A there is a minimal function f ∈ M \OA (generating a minimal clone) such that M = { f }∗(1).

Minimal functions are separated by Rosenberg’s Classiﬁcation Theorem [26] for minimal clones into ﬁve distinct categories ((II) is a special case of (V), but is often listed on its own).

Theorem 4 (See [26]). On a ﬁnite set A every minimal function f is of one of the following types: (1) (I) f ∈ OA and f ∈ Sym A is a permutation of prime order or a retractive operation f ◦ f = f 6= idA; (2) (II) f ∈ OA is idempotent, i.e., f ◦ (idA, idA) = idA; (3) (III) f ∈ OA is a ternary (minority) Mal’cev operation arising as f (x, y, z) := x + y + z for x, y, z ∈ A from a Boolean group hA; +i; (3) (IV) f ∈ OA is a ternary majority operation; (n) (V) f ∈ OA is a proper semiprojection of arity n where 3 ≤ n ≤ |A|.

In this paper, we address centralising monoids witnessed by a minimal function of types (I)–(III) with a special focus on the set {0, 1, 2, 3}; those relating to type (IV) have already been considered in [27], the ones of type (V) are part of ongoing research. Before we can tend to this problem in more detail, we need further background information on clones, such as the centraliser of a constant operation.

Lemma 5 (Lemma 1.9 in [10], Lemma 5.2 in [21]). For any n ∈ N and a ∈ A we have n (n)o∗ ca = PolA{{a}} = { f ∈ OA | a ∈ ﬁx( f )}.

(m) (n) Proof. Let m ∈ N and f ∈ OA . We have f ⊥ ca if and only if for all x11, ... , xmn ∈ A the condition

(n) (n) f (a,..., a) = f (ca (x11,..., x1n),..., ca (xm1,..., xmn)) (n) = ca ( f (x11,..., xm1),..., f (x1n,..., xmn)) = a

holds, that is, exactly if f (a,..., a) = a, i.e., precisely if a ∈ ﬁx( f ), or f ∈ PolA{{a}}.

(1)∗ ∗ Corollary 6 (Lemma 5.3 in [21]). The centraliser CA = CA is the clone of all idempotent operations.

∗ (1)∗ T n (1)o T Proof. We have CA = a∈A ca = a∈A PolA{{a}} = PolA{{a} | a ∈ A} due to Lemma5.

The following characterisation is also very well known. Symmetry 2021, 13, 1471 7 of 40

Lemma 7. For a clone F on any set A the following statements are equivalent:

(a) F is a clone of idempotent operations, i.e., F ⊆ PolA{{a} | a ∈ A}. (b) For every f ∈ F and all a ∈ A we have f (a,..., a) = a. (1) (c) F = {idA}.

Proof. Condition (b) simply spells out the preservation of every singleton set {a} in (a). (1) From this it follows that every f ∈ F satisﬁes f (a) = a for every a ∈ A, i.e., f = idA. For F contains all projections, (b) implies (c). Now since F is a clone, with every f ∈ F (1) also f ◦ (idA, ... , idA) ∈ F . If (c) holds, then f ◦ (idA, ... , idA) ∈ {idA}, and this shows statement (b).

(1)∗ Lemma 8. For any set A we have (JA ∪ CA) ∩ CA = JA.

Proof. For |A| < 2 this is trivially true. For |A| ≥ 2, no constant map is idempotent, so, (1)∗ by Corollary6, (JA ∪ CA) ∩ CA can only consist of projections. The converse inclusion is trivial.

When proving that a certain centralising monoid is maximal, it will be necessary to ∗∗(1) (1) compute the centralising monoid M generated by some set M ⊆ OA and to show that it is the full transformation monoid on A. To do this we shall always demonstrate the seemingly stronger condition that the centraliser of M is trivial. The following lemma shows that this is in fact a necessary step to take whenever |A| ≥ 3.

Lemma 9. For any set A with |A| ≥ 3 and any subset M ⊆ OA the following statements are equivalent: ∗∗(1) (1) (a) M = OA . ∗∗ (b) M = OA. ∗ (c) M = JA.

Proof. The implications (c) =⇒ (b) =⇒ (a) are obvious as every function commutes with (1) ∗∗ ∗ (1)∗ projections. Now, if (a) holds, then OA ⊆ M , which is equivalent to F := M ⊆ OA . By Lemma1, this means the functions in F preserve the kernel of any unary operation, so they preserve every equivalence relation on A. Therefore, F ⊆ PolA Eq(A), and for |A| ≥ 3 the latter clone equals JA ∪ CA as is shown in Example 3.3 of [30] (p. 136). Now (1)∗ (1)∗ (1)∗ F ⊆ OA ⊆ CA implies that F ⊆ (JA ∪ CA) ∩ CA , so Lemma8 shows that (c) must be satisﬁed.

Note that for |A| = 2, Lemma9 fails as, e.g., the clone L of all afﬁne linear functions, which is the centraliser of the (ternary minority) Mal’cev function on the two-element set, ∗∗ ∗∗(1) (1) (1) contains all unary operations. So L = L ( OA, but L = L = OA . To establish for a subset M ⊆ OA condition (c) of Lemma9 or to prove at least ∗(1) M = {idA}, we shall exploit a theorem of Länger and Pöschel [30] on (strongly) constantively rigid systems of (binary) relations.

Proposition 10 (cf. [30] (Theorem 2.13, p. 136, Theorem 2.3, p. 133)). Let A be any (finite (2) or infinite) set with |A| ≥ 4, F ⊆ OA and let Q ⊆ InvA F satisfy the following two separation conditions (see Definition 2.7, p. 134 et seq. in [30]) (B) for all pairwise distinct x, y, z ∈ A there is $ ∈ Q such that (x, y) ∈ $ 63 (x, z); (D) for all pairwise distinct x, y, z, u ∈ A there is $ ∈ Q such that (x, y) ∈ $ 63 (z, u). (1) (1) Then F ⊆ {idA} ∪ CA ; moreover, if all relations in Q are reflexive, then F ⊆ JA ∪ CA. Symmetry 2021, 13, 1471 8 of 40

(1) (1) Proof. By our assumption we have F ⊆ PolA Q and, in particular, F ⊆ PolA Q. We (1) use Theorem 2.13 of [30] to infer that Q is constantively rigid, i.e., PolA Q ⊆ JA ∪ CA, (1) (1) wherefore F ⊆ {idA} ∪ CA . If, additionally, Q only contains reﬂexive relations, then Theorem 2.3 of [30] states that constantive rigidity of Q is equivalent to strong constantive rigidity, that is, F ⊆ PolA Q ⊆ JA ∪ CA.

Remark 11. There are two more conditions, (A) and (C), in [30], which can also be employed to ensure (strong) constantive rigidity of systems Q of (binary reﬂexive) relations, see, e.g., Theorems 2.11 and 2.12 of [30]. In particular, via Theorem 2.12, (B) and (C) together may be used to obtain (strong) constantive rigidity also when |A| = 3, and (C) automatically follows from (B) if all the relations in Q are symmetric (see Lemma 2.8 in [30]), for example, equivalence relations. More concrete constantively rigid systems of relations are provided in Section 3 of [30], some of them also being strongly constantively rigid.

Corollary 12. Let A be any set with |A| ≥ 4 and M ⊆ OA be such that for each a ∈ A there is ( ) s ∈ M satisfying a ∈/ fix(s). Moreover assume that Q ⊆ [M•] 2 satisfies conditions (B) and RA ∗(1) ∗ (D), then M = {idA}; if all the relations in Q are reflexive, then M = JA; if, moreover, the relations used to satisfy (B) are reflexive and symmetric, then the result also follows for |A| = 3.

Proof. Let F := M∗ in Proposition 10 and note that [M•] ⊆ Inv Pol M• = Inv M∗. RA A A A ∗(1) (1) ∗ So Proposition 10 entails that M = F ⊆ JA ∪ CA and M = F ⊆ JA ∪ CA under the reflexive assumption. If the relations used in (B) are reflexive and symmetric, we combine the reasoning of Proposition 10 with Remark 11 to get F ⊆ JA ∪ CA also for |A| = 3. (n) ∗ Finally, F ∩ CA = ∅, for the supposition of ca ∈ F = M for some a ∈ A and n ∈ N n (n)o∗ would imply M ⊆ ca = { f ∈ OA | a ∈ fix( f )}, see Lemma5, which contradicts the assumption on M.

Corollary 13. Let A be any set with |A| ≥ 3 and M ⊆ OA be such that for each a ∈ A there is s ∈ M satisfying a ∈/ ﬁx(s). Moreover assume that (B’) for all pairwise distinct x, y, z ∈ A there is s ∈ M(1) such that s(y) = s(x) 6= s(z), (D’) for all pairwise distinct x, y, z, u ∈ A there is s ∈ M(1) such that s(x) = s(y) and s(z) 6= s(u), ∗ then M = JA. n o ( ) ∈ (1) ⊆ (2) ∗ Proof. Use Corollary 12 on the equivalence relations ker s s M InvA M .

Corollary 14. Let A be any set with |A| ≥ 4 and M ⊆ OA be such that for each a ∈ A there is s ∈ M satisfying a ∈/ ﬁx(s). Moreover assume that (B”) for all pairwise distinct x, y, z ∈ A there is s ∈ M(1) such that s(y) = s(x) 6= s(z) or s(x) = y or s(y) = x 6= s(z), (D”) for all pairwise distinct x, y, z, u ∈ A there is s ∈ M(1) satisfying one of the conditions s(x) = s(y) ∧ s(z) 6= s(u) or s(x) = y ∧ s(z) 6= u or s(y) = x ∧ s(u) 6= z, ∗(1) then M = {idA}. n o n o = ( ) ∈ (1) ∪ (1)• ∪ •−1 ∈ (1) ⊆ (2) ∗ Proof. Use the set Q : ker s s M M s s M InvA M in Corollary 12.

3. Monoids Witnessed by Unary Operations There are two types of unary minimal functions in Rosenberg’s Theorem4. The ﬁrst are permutations of prime order, that is, their cycle structure consists only of ﬁxed points and cycles of length p for some prime p. The second are idempotent or retractive Symmetry 2021, 13, 1471 9 of 40

(1) operations f ∈ OA , which satisfy f ◦ f = f 6= idA. These are exactly those non-identical unary operations which ﬁx every point of their image.

3.1. Computational Results for {0, 1, 2, 3} We started by computing a commutation table of all 256 unary operations on the (1) (1) set A = {0, 1, 2, 3}, i.e., the formal context K1 = OA , OA , ⊥ in the language of [35,36]. Being a 256 × 256 Boolean matrix this table is already confusingly big and we therefore do not present it here. For purposes of veriﬁcation, in Listing1 we have instead added simple Python code that allows anybody to reproduce such a table if desired. Us- ing standard algorithms presented in Section 2.1 of [35] or Chapter 2 of [36], one may compute from K1 that there are exactly 1485 centralising monoids on {0, 1, 2, 3} that can be witnessed by sets of unary operations. However, we are only interested in the co- atoms of this large lattice. By virtue of the Galois connection represented by K1, every co-atom M is the Galois derivative N∗(1) of an atom N on the dual side of the Galois connection. Since in the lattice of closed sets of any closure operator the atoms must be ∗(1)∗(1) (1) singly generated, it follows that N = { f } for a single unary function f ∈ OA and so M = N∗(1) = { f }∗(1)∗(1)∗(1) = { f }∗(1). Hence, to obtain the co-atoms, we only need to (1) iterate over the 255 rows of K1 belonging to each f ∈ OA \ {idA} (or just the unary f from Rosenberg’s Theorem) and check, which { f }∗(1) are maximal under set inclusion. The result is that there are 49 co-atoms in the lattice of closed sets of K1, and subsequently we shall explain our computational ﬁndings in terms of the types of unary operations from Rosenberg’s Theorem, i.e., permutations of prime order and non-identical retractions. In the following two subsections we shall then give theoretical evidence why the 49 co-atoms arise on {0, 1, 2, 3}. It is worth noting that these 49 co-atoms in the lattice belonging to K1 are merely candidates for maximal centralising monoids, and we shall indeed demonstrate that some co-atoms of K1 fail to be maximal among all centralising (1) monoids, that is, are no co-atoms with respect to OA, OA , ⊥ .

Listing 1. Python code to print a commutation table (0-1-matrix) for all 256 unary operations f on {0, 1, 2, 3}, where rows and columns are enumerated in lectic order of the value tables f ◦ (0, 1, 2, 3) from (0, 0, 0, 0), (0, 0, 0, 1), (0, 0, 0, 2),..., (3, 3, 3, 3). #!/usr/bin/env python f = [0] * 4 g = [0] * 4 row = [0] * 256 separator = "␣" # could also be an empty string or ", " if desired f o r f[0] i n range (4): f o r f[1] i n range (4): f o r f[2] i n range (4): f o r f[3] i n range (4): j=0 f o r g[0] i n range (4): f o r g[1] i n range (4): f o r g[2] i n range (4): f o r g[3] i n range (4): fgcommute= True f o r x i n range (4): fgcommute= fgcommute and f[g[x]] ==g[f[x]] i f (fgcommute): row[j] = 1 # indicates commutation of f and g e l s e : row[j] = 0 # indicates non-commutation of f and g j=j+1 p r i n t (separator.join( map( s t r ,row))) # output table row Symmetry 2021, 13, 1471 10 of 40

On a four-element set there are three types of permutations of prime order: three- cycles, transpositions or products of two disjoint transpositions. Up to conjugacy these can be represented by the following three functions on {0, 1, 2, 3}: (0, 1, 2), (0, 1), (0, 1)(2, 3). 4 There are 2 · (1) = 8 three-cycles (4 choices of the unique fixed point and 2 for the image of the first cycle element), but it will turn out that centralising monoids witnessed by these operations are not maximal (Lemma 16), not even with respect to K1. However, the monoids described by the other prime permutations are maximal when considered among the monoids witnessed by only unary operations (i.e., the closure system of K1). Yet again, they will not be maximal in the lattice of all centralising monoids on {0, 1, 2, 3}, 4 see Corollary 22. We have (2) = 6 transpositions (choosing the two transposed elements) 1 4 and 2 · (2) = 3 products of disjoint transpositions since such a function can be obtained by choosing the first transposed pair or the second one. Altogether there are 9 prime permutations (of order 2) that describe maximal monoids of K1. The non-trivial retractive functions on a four-element set can be separated into the following types: being constant (one-element image, there are 4 such functions), having a 4 3 two-element image, or having a three-element image (there are (3) · (1) = 12 such functions, first choosing the image and then the value of the element outside the image within it). The retractions with a two-element image can moreover be split up into those mapping the two elements outside the image to the same value and those mapping them to both distinct 4 values of the image. In both subcases we have (2) · 2 = 12 functions as we need to choose the two-element image and then one of two ways how the elements outside the image can be mapped into it. Retractions of all four types witness centralising monoids that are maximal among those centralising monoids witnessed by unary functions, see Corollary 27, and there are 3 · 12 + 4 = 40 non-trivial retractions on {0, 1, 2, 3}. Moreover, it is known n (1)o∗(1) from Theorem 5.1 of [21] that ca with a ∈ A is a maximal centralising monoid in general (that is on any finite set A with |A| ≥ 3 in the lattice of all centralising monoids), (1) so the 4 centralising monoids of constants are co-atoms with respect to OA, OA , ⊥ , not just K1. In total, we have 40 + 9 = 49 centralising monoids on {0, 1, 2, 3} that are maximal in the 1485-element lattice of those monoids that can be witnessed by sets of unary operations. ∗( ) These can be grouped together into 6 conjugacy types that can be represented as {(0, 1)} 1 , ∗(1) ∗(1) n (1)o ∗(1) ∗(1) ∗(1) {(0, 1)(2, 3)} , c0 , { f24} , { f16} and { f17} where the retractions f24, f16 and f17 are given as f24 ◦ (0, 1, 2, 3) = (0, 1, 2, 0), f16 ◦ (0, 1, 2, 3) = (0, 1, 0, 0) and f17 ◦ (0, 1, 2, 3) = (0, 1, 0, 1). From the explicit calculations (and the characterisations in the following subsections) it follows that

∗(1) ∗(1) {(0, 1)} = 16 {(0, 1)(2, 3)} = 16 ∗(1) n (1)o ∗(1) c = 64 { f } = 36 0 24

∗(1) ∗(1) { f16} = 20 { f17} = 16.

∗(1) As (0, 1)(2, 3) has no ﬁxed points but (0, 1) and f17 have, the monoid {(0, 1)(2, 3)} ∗(1) ∗(1) does not contain constant operations (cf. Lemma5) but {(0, 1)} and { f17} do. There- D ∗(1) E ∼ D ∗(1) E ∼ D ∗(1) E fore, {(0, 1)} ; ◦ 6= {(0, 1)(2, 3)} ; ◦ 6= { f17} ; ◦ . Furthermore, computa- D ∗(1) E D ∗(1) E tions show that {(0, 1)} ; ◦ has 96 binary term functions, but { f17} ; ◦ has 262 of D ∗(1) E ∼ D ∗(1) E them, so {(0, 1)} ; ◦ 6= { f17} ; ◦ . Therefore, the co-atoms of the lattice of intents of K1 fall into exactly 6 distinct isomorphism classes of monoids. Moreover, we computed that all 1485 monoids in the lattice of intents of K1 can be separated into 106 classes up to element-wise conjugacy (cf. Section4 for more explanation). Symmetry 2021, 13, 1471 11 of 40

3.2. Monoids Witnessed by Permutations In Lemma 4.11 of [27] (see also Lemma 4.10 of [37]) we provided a characterisation of when a finitary operation f ∈ OA on a finite set A commutes with a permutation s ∈ Sym A, based on examining orbits of the permutation group generated by s. Under the assumption that f is a function the given condition was sufficient to ensure that f ⊥ s, but blindly fulfilling the orbit condition could also lead to some value assignments for f that would contradict the assumption of f being a function. Thus, some care had to be taken when working with Lemma 4.11 from [27], but this was never much of an issue when studying majority operations f on small sets as in [27]. Here we improve the mentioned characterisation by placing an additional necessary condition on the choice of the function values such that no contradictions can occur.

Lemma 15. For a finite set A, n ∈ N+ and s ∈ Sym A let S := h{s}iSym A be the permutation group generated by s and T ⊆ An be a transversal of the orbits of the action of S on An via (n) (s˜, x) 7→ s˜ ◦ x. For f ∈ OA we have f ⊥ s if and only if for every x ∈ T the length of the orbit of f (x) under S divides the length ` of the orbit of x under S and f (sj ◦ x) = sj( f (x)) for all exponents j satisfying 1 ≤ j < `. In particular if for all x ∈ T a value f (x) is chosen such that the size of the orbit of f (x) is a divisor of the size ` of the orbit of x, then there is a unique extension of this partial definition to an n-ary function f ∈ {s}∗ by defining f (sj ◦ x) := sj( f (x)) for all 1 ≤ j < ` and x ∈ T.

Proof. Let m := |S| be the order of s. Lemma 4.11 of [27] shows that f ⊥ s if and only if for all x ∈ T the condition f (sj ◦ x) = sj( f (x)) holds for all 0 ≤ j < m. The length ` of the orbit of x is the least common multiple of the lengths of the orbits of the entries of x and each of these lengths divides m, so ` divides m. So f (sj ◦ x) = sj( f (x)) for all 1 ≤ j < ` follows from f ⊥ s. If t is the size of the orbit of f (x), then this orbit is sj( f (x)) 0 ≤ j < t and st( f (x)) = f (x). Writing ` = q · t + r with 0 ≤ r < t, it follows from the deﬁnition of ` and the commutation condition that

f (x) = f (s` ◦ x) = f (s ◦ s`−1 ◦ x) = s( f (s`−1 ◦ x)) = s(s`−1( f (x))) = s`( f (x)) = sr( f (x)),

= j( ( )) ⊥ so r 0 as the elements s f x 0≤j

From Lemma 15 it follows again for every n-ary f ∈ {s}∗ that if s ◦ x = x, that is, x ∈ (fix(s))n, then f (x) must belong to an orbit of size 1. Hence, f (x) ∈ fix(s) and f ∈ PolA{fix(s)}. So we obtain once more one of the necessary conditions for functions in the centraliser that were given in Lemma1. Symmetry 2021, 13, 1471 12 of 40

We can easily adapt an argument of Harnau’s [10] (Lemma 2.9, p. 345), given for centraliser clones of unary operations, to centralising monoids. This shows that on a four-element set no three-cycle can witness a maximal centralising monoid (nor a maximal centraliser clone, by Harnau’s result).

∗(1) Lemma 16. Let A be a set with 2 ≤ |A| < ℵ0, and s ∈ Sym A be a permutation such that {s} is a maximal centralising monoid, then the number of ﬁxed points of s is not 1.

Proof. Assume for a contradiction that s ∈ Sym A witnesses a maximal centralising monoid and has a ∈ A as its unique fixed point. Since {s}∗(1) is maximal, we have p s = idA 6= s for some prime p. Applying the above Lemmas1 and5 we then conclude that ∗ ∗(1) ∗ n (1)o ∗(1) n (1)o (1) {s} ⊆ PolA{{a}} = ca , so {s} ⊆ ca ( OA due to |A| ≥ 2. As s 6= idA, its cycle representation has at least one p-cycle (a1, ... , ap), and a ∈/ a1,..., ap because (1) n (1)o∗(1) it is fixed by s. We define f ∈ PolA {{a}} = ca by f (a1) := a1 and f (x) := a for all x ∈ A \ {a1}. If f were to commute with s, then Lemma 15 (or the definition) would imply that a = f (a2) = f (s(a1)) = s( f (a1)) = s(a1) = a2 contradicting a ∈/ a1,..., ap . ∗(1) ∗(1) n (1)o ∗ ∗(1) n (1)o (1) ∗(1) Hence, f ∈ ca \ {s} , so {s} ( ca ( OA and {s} cannot be a maximal centralising monoid (not even maximal among just those witnessed by unary operations).

The following lemma describes (for the case when ` is a prime) the centralising monoids witnessed by permutations of prime order.

Lemma 17. Let s ∈ Sym A be a permutation on a ﬁnite set A that is a product of t ≥ 0 disjoint cycles of length ` ≥ 2 and has |A| − t · ` ﬁxed points. That is, s is of the form i i i i0 0 0 0 0 s = ∏0≤i

Proof. We simplify the condition given in Lemma 15. A transversal of the orbits of the i cyclic permutation group generated by s is given by T = a0 0 ≤ i < t ∪ fix(s). The condition in Lemma 15 expressed for those x ∈ T that are fixed by s is exactly equivalent to f ∈ PolA{fix(s)} since the only positive divisor of 1 is 1. For all x ∈ T \ fix(s), i.e., i x = a0 for some 0 ≤ i < t, the length of the orbit generated by x is `, so the divisibility i condition in Lemma 15 is fulfilled for any choice of the value f (x) = f (a0) ∈ A as the only orbit sizes are 1 or `. The remaining part of the condition translates to the equality of i j i j j j i f (aj) = f (s (a0)) = f (s (x)) and s ( f (x)) = s ( f (a0)) for all 1 ≤ j < `.

∗(1) (1) When |A| ≥ 2 and t ≥ 1, that is, s 6= idA, it is easy to see that {s} ( OA , because (1) ∗(1) 0 0 we can deﬁne f ∈ OA \ {s} with f (a1) 6= s( f (a0)), thereby violating condition (b) of Lemma 17. The following corollary is also evident.

Corollary 18. Let A ⊇ {0, 1} be a ﬁnite set and s := (0, 1) be a transposition. Then {s}∗(1) equals n o ∈ (1){ \ { }} ◦ ( ) ∈ {( ) ( )} ∪ { ( ) | ∈ \ { }} f PolA A 0, 1 f 0, 1 0, 1 , 1, 0 a, a a A 0, 1 ;

moreover, for A = {0, 1, 2, 3} we have n o {( )}∗(1) = ∈ (1){{ }} ◦ ( ) ∈ {( ) ( )} ∪ { ( ) | ∈ { }} 0, 1 f PolA 2, 3 f 0, 1 0, 1 , 1, 0 a, a a 2, 3 . Symmetry 2021, 13, 1471 13 of 40

Next we prove that the centraliser of any proper supermonoid of a centralising monoid witnessed by a ﬁxed-point free product of disjoint transpositions or a single transposition with at least two ﬁxed points is idempotent. Hence, by Lemma7, the centralising monoid of such a transposition is maximal in the lattice of all centralising monoids witnessed by unary operations (the intent lattice of K1).

Lemma 19. For a permutation s ∈ Sym A on A with 4 ≤ |A| < ℵ0 of order two that can be written as a ﬁxed-point free product s = (a1, a2)(a3, a4) ··· (ak−1, ak) of disjoint transpositions ∗(1) ∗ where A = {a1,..., ak} and any transformation monoid M ) {s} , the centraliser clone M ∗(1) ∗(1)∗(1) contains only idempotent operations, i.e., M = {idA}. Hence, if M = M , then (1) M = OA .

∗(1) ∗(1) Proof. We use Corollary 14 to show that M = {idA} for every monoid M ) {s} . ∗( ) Note that s has only two-cycles, so fix(s) = ∅ and {s} 1 is fully described by condition (b) of Lemma 17. Transversals of the orbits are given by picking exactly one element from each cycle, and any assignment of values to (some of) the elements of the transversal can be completed to a function g ∈ {s}∗(1) ⊆ M, cf. Lemmas 17 and 15. In particular, for every a ∈ A we can pick b ∈ A \ {a} and the partial definition g(a) := b 6= a can be extended to some g ∈ {s}∗(1) ⊆ M with a ∈/ fix(g). Hence, the initial assumption of Corollary 14 is satisfied. To show condition (B”), take distinct x, y, z ∈ A and extend the partial definition g(x) := y 6= z to some g ∈ {s}∗(1) ⊆ M. For condition (D”) consider distinct x, y, z, u ∈ A and some h ∈ M \ {s}∗. Since h does not commute with s there is a ∈ A such that for b := s(a), p := h(a) and q := h(b) the inequality q = h(b) = h(s(a)) 6= s(h(a)) = s(p) holds. We can define some g ∈ {s}∗(1) with g(p) = g(q) = y. Namely, if p = q, this is clear, and if q 6= p, then q ∈/ {p, s(p)} and thus belongs to a different orbit than p, wherefore the assignment of y on a transversal containing {p, q} can be extended to some g ∈ {s}∗(1). Moreover, we define g˜(x) := a and g˜(z) := b. If z = s(x), then this assignment is compatible with condition (b) of Lemma 17 because g˜(s(x)) = g˜(z) = b = s(a) = s(g˜(x)), so it can be extended to some g˜ ∈ {s}∗(1). If z 6= s(x), then z ∈/ {x, s(x)} and thus belongs to a distinct orbit than x, wherefore the given assignment on a transversal including {x, z} can be extended to some g˜ ∈ {s}∗(1). As a consequence, we have g ◦ h ◦ g˜ ∈ M, and it satisfies g(h(g˜(x))) = g(h(a)) = g(p) = y ∗(1) and g(h(g˜(z))) = g(h(b)) = g(q) = y 6= u. Invoking Corollary 14, M = {idA}, and by Lemma7, M∗ has only idempotent operations.

Lemma 20. For any ﬁnite set A ⊇ {0, 1, 2, 3}, a transposition f = (0, 1) and a transforma- ∗(1) ∗ tion monoid M ) { f } , the centraliser clone M consists only of idempotent operations, i.e., ∗ ∗(1) ∗(1)∗(1) (1) M ⊆ PolA{{a} | a ∈ A}. Hence, M = {idA}, and if M = M , then M = OA .

Proof. We have to show that all functions in M∗ preserve all singletons of A. For every n o∗ (1) ∗(1) ∗ (1) a ∈ fix( f ) we have ca ∈ { f } ⊆ M by Lemma5, so M ⊆ ca a ∈ fix( f ) . Thus, again by Lemma5, we get M ⊆ PolA{ {a} | a ∈ fix( f )}. It only remains to show that ∗ ∗(1) M ⊆ PolA{{0}, {1}}, and for this task we shall exploit that there is some h ∈ M \ { f } . Subsequently we distinguish cases how h might look like according to Corollary 18. The first case is that h ∈/ PolA{fix( f )}, i.e., there is some t ∈/ {0, 1} such that h(t) ∈ {0, 1}. n (1)o ∗(1) n (1) (1)o n (1) (1)o By the above, f , ct ⊆ { f } ⊆ M, and then h ◦ ct , f ◦ h ◦ ct = c0 , c1 ⊆ M. ∗ ∗ n (1) (1)o Again by Lemma5 we infer M ⊆ c0 , c1 = PolA{{0}, {1}}. From now on assume that h ∈ PolA{A \ {0, 1}}. The second case is that one of h(0), h(1) belongs to fix( f ), but h(0) 6= h(1). If h(1) ∈ fix( f ), then we can use h ◦ f ∈ M instead, where h( f (0)) = h(1) is in fix( f ) and still h( f (0)) = h(1) 6= h(0) = h( f (1)). Therefore, no generality is lost in assuming that h(1) 6= h(0) =: a ∈ fix( f ). Symmetry 2021, 13, 1471 14 of 40

As a subcase we assume that h(b) = h(0) = a for all b ∈ fix( f ). By Lemma 17 there is g ∈ { f }∗(1) with g(h(0)) = g(a) = a and g(x) = t for all x ∈ A \ {a} where (1) t ∈ fix( f ) \ {a}. Such a t exists as |A| ≥ 4; in particular g(h(1)) = t. Since ct ∈ M, (1) • the singleton {t} = im ct is primitive positively definable from M (see Lemma1), therefore so is

(g ◦ h)−1[{t}] = { x ∈ A | ∃y ∈ A : g ◦ h(x) = y ∧ y ∈ {t}} n o (1) = x ∈ A ∃y, z ∈ A : g ◦ h(x) = y ∧ ct (z) = y = {1}.

−1 • n (1) o Likewise, we can define (g ◦ h ◦ f ) [{t}] = {0} from M due to ct , f , g, h ⊆ M. Consequently, every function in M∗ preserves {0} and {1}. The opposite subcase is that there is b ∈ fix( f ) with h(b) 6= h(0) = a ∈ fix( f ). The (1) function g1 ∈ OA defined via g1(x) = x for x ∈ {0, 1} and g1(x) = b for x ∈ fix( f ) ∗(1) (1) belongs to { f } by Corollary 18. Take again t ∈ fix( f ) \ {a} and define g2 ∈ OA by ∗(1) g2(a) = a and g2(x) = t for all x ∈ A \ {a}; by Lemma 17, g2 ∈ { f } . As h(1) and h(b) are distinct from a, it follows that g2 ◦ h ◦ g1(0) = a and g2 ◦ h ◦ g1(x) = t for all x ∈ A \ {0}. Likewise g2 ◦ h ◦ g1 ◦ f (1) = a and g2 ◦ h ◦ g1 ◦ f (x) = t for all x ∈ A \ {1}. −1 −1 Therefore, (g2 ◦ h ◦ g1) [{a}] = {0} and (g2 ◦ h ◦ g1 ◦ f ) [{a}] = {1}, and as above these • n (1)o preimages are primitive positively definable from M as g1, g2, h, f , ca ⊆ M. Hence, ∗ M ⊆ PolA{{0}, {1}}. This finishes the second case where h(0) 6= h(1) and one of them is fixed by f . ∗ The remaining case is when h ∈ PolA{fix( f )} \ { f } , but the second case does not take place, i.e., none of h(0), h(1) is fixed by f . This implies {h(0), h(1)} ⊆ {0, 1} and, moreover h(0) = h(1), because h ∈/ { f }∗, see Corollary 18. Let us assume that h(0) = h(1) = 0, for otherwise, we might consider f ◦ h ∈ M, which operates identically to h on fix( f ) and has swapped values compared to h on {0, 1}. Certainly, 0 ∈ fix(h). If it is the only fixed point, then fix(h) = {0} and fix( f ◦ h) = {1}. By Lemma1 both of these sets are primitive • ∗ positively definable from M since f , h ∈ M. Therefore, M ⊆ PolA{{0}, {1}}. Now let us assume that there is a ∈ A \ {0, 1} = fix( f ) such that {0, a} ⊆ fix(h) (1) and let b ∈ fix( f ) \ {a}. By Lemma 17, the functions ga, gb ∈ OA , defined by gz(x) = x ∗(1) for x ∈ {0, 1} and gz(x) = z for x ∈ fix( f ) and both z ∈ {a, b}, belong to { f } ⊆ M. We have h ◦ ga(x) = 0 for x ∈ {0, 1} and h ◦ ga(x) = h(a) = a for x ∈ fix( f ) since a ∈ fix(h). Therefore fix(h ◦ ga) = {0, a}. Likewise, gb ◦ h ◦ ga(x) = 0 for x ∈ {0, 1} and gb ◦ h ◦ ga(x) = gb(a) = b for x ∈ fix( f ) since a ∈ fix( f ). Thus, fix(gb ◦ h ◦ ga) = {0, b}. As • h ◦ ga, gb ◦ h ◦ ga ∈ M, these sets of fixed points are primitive positively definable from M (Lemma1), and so is their intersection {0} = {0, b} ∩ {0, a}, where we use that b 6= a by the choice of b. Similarly, fix( f ◦ h ◦ ga) = {1, a} and fix( f ◦ gb ◦ h ◦ ga) = {1, b}, and hence {1} = {1, a} ∩ {1, b} is primitive positively definable from M•, as well. We conclude that ∗ M ⊆ PolA{{0}, {1}}.

Lemma 21. For any ﬁnite set A ⊇ {0, 1, 2} and the transposition s = (0, 1), let a binary operation ( ) b ∈ O 2 be deﬁned by b(x, y) := s(y) if x ∈ A \ {0, 1}, b(x, y) := y if y ∈ A \ {0, 1} and A n o ∗(1) ∗(1) ˙ (1) (1) b(x, y) := x if x, y ∈ {0, 1}. Then b is idempotent and {b} = {s} ∪ c0 , c1 .

Proof. We first note that b is well defined since A \ {0, 1} = fix(s) and thus for any x, y ∈ fix(s) we have s(y) = y = b(x, y). In particular, b(y, y) = y for y ∈ fix(s), and b(y, y) = y for y ∈ {0, 1}, too. Hence, b(y, y) = y for all y ∈ A, so b is idempotent. (1)∗ (1) ∗(1) As b is idempotent, Corollary6 yields that b ∈ CA , i.e., CA ⊆ {b} . So, first, we take any f ∈ {s}∗(1) and prove that it commutes with b. For this we consider any x, y ∈ A and want to show that b( f (x), f (y)) = f (b(x, y)). This is clear from idem- Symmetry 2021, 13, 1471 15 of 40

potence of b if x = y; thus, we distinguish three more cases re x 6= y. Assume first that f (x) ∈ fix(s); hence b( f (x), f (y)) = s( f (y)) = f (s(y)). Now if also x ∈ fix(s), then b(x, y) = s(y) and so f (b(x, y)) = f (s(y)) = b( f (x), f (y)). Otherwise, x ∈ {0, 1} and we distinguish two subcases. If y ∈ fix(s), then b(x, y) = y, wherefore we get b( f (x), f (y)) = f (s(y)) = f (y) = f (b(x, y)). Alternatively, x, y ∈ {0, 1} and b(x, y) = x. Since x 6= y, we have s(y) = x, and so b( f (x), f (y)) = f (s(y)) = f (x) = f (b(x, y)). Our second case is that f (y) ∈ fix(s), whence b( f (x), f (y)) = f (y). Clearly, if also y ∈ fix(s), then f (b(x, y)) = f (y) = b( f (x), f (y)). Hence, suppose that y ∈ {0, 1}. We get now that b( f (x), f (y)) = f (y) = s( f (y)) = f (s(y)), using that f (y) ∈ fix(s) and f ∈ {s}∗(1). If x ∈ fix(s), then b(x, y) = s(y) and b( f (x), f (y)) = f (s(y)) = f (b(x, y)). Finally, the subcase x, y ∈ {0, 1} with x 6= y remains, which is treated as shown at the end of the previous paragraph. The third and last case is that f (x), f (y) ∈ A \ fix(s) = {0, 1}. Then x, y must belong ∗(1) to {0, 1} as f ∈ {s} ⊆ PolA{fix(s)} by Lemma 17. From the definition of b we now infer b(x, y) = x and also b( f (x), f (y)) = f (x) = f (b(x, y)). For the opposite inclusion we consider any f ∈ {b}∗(1) that is not constant and have ∗(1) (1) to show f ∈ {s} . First we demonstrate that f ∈ PolA {fix(s)}. Taking any y ∈ fix(s), we have b(x, y) = y. Since f ∈/ CA, there is x ∈ A such that f (x) 6= f (y). Now the commutation condition with respect to b gives b( f (x), f (y)) = f (b(x, y)) = f (y). Because f (x) 6= f (y), this is not possible if f (y) ∈ {0, 1}, i.e., we obtain f (y) ∈ fix(s). Knowing now that f ∈ PolA{fix(s)} we consider any y ∈ A and with it 2 ∈ fix(s), whence also f (2) ∈ fix(s). The definition of b implies s( f (y)) = b( f (2), f (y)) and b(2, y) = s(y), wherefore s( f (y)) = b( f (2), f (y)) = f (b(2, y)) = f (s(y)) because we ∗ ∗ (1) ∗(1) have f ∈ {b} . This shows that f ∈ {s} as desired. Finally, if f = ca ∈ {b} and n (1)o∗(1) (1) a ∈/ {0, 1}, then a ∈ fix(s) and so Lemma5 shows s ∈ ca , i.e., ca ⊥ s.

Corollary 22. Assume 4 ≤ |A| < ℵ0 and let s ∈ Sym A be a transposition or a product of disjoint transpositions without ﬁxed points. Then {s}∗(1) is a co-atom in the lattice of all centralising monoids on A witnessed by unary operations. However, the centralising monoid of a transposition is never a maximal centralising monoid, and the centralising monoid of a product of two disjoint transpositions fails to be maximal in the case where |A| = 4.

Proof. Maximality in the lattice of centralising monoids witnessed by unary operations (that is with respect to the closure operator X 7→ X∗(1)∗(1)) follows from Lemmas 19 and 20. That transpositions do not yield co-atoms in the lattice of all centralising monoids is a consequence of Lemma 21. Finally, we consider a product s of two disjoint transpositions on A = {0, 1, 2, 3}, which we can represent up to conjugacy as s = (03)(12). If we 3 3−j represent unary operations u on {0, 1, 2, 3} as integer numbers ∑j=0 u(j)4 , then we have {s}∗(1) = {15, 27, 39, 51, 78, 90, 102, 114, 141, 153, 165, 177, 204, 216, 228, 240}. It can be seen from Table 4 in [27] that the centralising monoid witnessed by the majority operation f20 (see Table 1 in [27]) contains all these unary operations and, for example, all unary constants. { }∗(1) ∪ (1) { }∗(1) O(1) = ( )( ) Therefore, we have s C{0,1,2,3} ( f20 ( A , whence s 03 12 does not witness a maximal centralising monoid on {0, 1, 2, 3}.

3.3. Monoids Witnessed by Retractive Operations We now turn our attention from permutations of prime order to unary retractive (1) operations f ∈ OA (also called idempotent unary operations), that is, such f : A −→ A (1) where f ◦ f = f . This property for f ∈ OA is equivalent to f |im( f ) = idA|im( f ), wherefore (1) ﬁx( f ) = im( f ) for any retraction f ∈ OA . It follows that the only surjective retraction (1) (1) f ∈ OA is the one ﬁxing every point of A, so f = idA. In other words, if f ∈ OA \ {idA} Symmetry 2021, 13, 1471 16 of 40

is a non-identical retraction, then the set of ﬁxed points (image) ﬁx( f ) is a proper subset of A.

(1) (n) Lemma 23. Let A be any set, n ∈ N+ and f ∈ OA with f ◦ f = f . For h ∈ OA we n n have h ⊥ f if and only if h ∈ PolA{fix( f )} and for all x ∈ A \ (fix( f )) the condition h(x) ∈ f −1[{h( f ◦ x)}] is satisfied. In particular, since for every x ∈ An the tuple f ◦ x ∈ (im( f ))n = (fix( f ))n, for any choice of values h(x) ∈ fix( f ) for x ∈ (fix( f ))n there are (generally not unique) extensions h ∈ { f }∗(n).

Proof. For every individual x ∈ An the commutation condition f (h(x)) = h( f ◦ x) can equivalently be restated as h(x) ∈ f −1[{h( f ◦ x)}]. For all x ∈ An \ (fix( f ))n we use the reformulation, while for all x ∈ (fix( f ))n we keep the original condition, which simplifies to f (h(x)) = h(x) as f ◦ x = x. However, this is just expressing that h(x) ∈ fix( f ) for every n x ∈ (fix( f )) , i.e., that h ∈ PolA{fix( f )}. For the second part of the lemma, we observe that once values h(x) ∈ fix( f ) are selected for each tuple x ∈ (fix( f ))n, the condition h(x) ∈ f −1[{h( f ◦ x)}] determines the values on all remaining tuples since f ◦ x ∈ (fix( f ))n. The choices of values of h in the preimages can be made independently of each other without leading to contradictions, i.e., all possible choices for h described above indeed produce a function h ∈ { f }∗.

(1) For any retraction f ∈ OA and any element x ∈ ﬁx( f ) = im( f ), we can write −1 f [{x}] = {x} ∪ Ax where Ax ⊆ A \ ﬁx( f ) is possibly empty. Every Ax 6= ∅ adds a requirement to the commutation condition, and (for unary operations) it is useful to rephrase Lemma 23 in terms of these sets.

(1) −1 Corollary 24. For a retraction f ∈ OA let I := x ∈ ﬁx( f ) Ax := f [{x}] \ {x} 6= ∅ . Then ( ) (h(j) =: i ∈ I ∧ h(a) ∈ {i} ∪ A ) ∨ { }∗(1) = ∈ (1){ ( )} ∀ ∈ ∀ ∈ i f h PolA ﬁx f j I a Aj : . (h(j) ∈/ I ∧ h(a) = h(j))

Proof. We use Lemma 23 for n = 1 and note that for j ∈ I and a ∈ Aj we have f (a) = j −1 −1 and h( f (a)) = h(j) =: i ∈ ﬁx( f ). So f [{h( f (a))}] = f [{i}] = {i} ∪ Ai, which equals just {i} = {h(j)} whenever i ∈ ﬁx( f ) \ I.

∗(1) (1) (1) It is evident that we have { f } ( OA for any retraction f ∈ OA \ {idA}. Namely, since f 6= idA, we have fix( f ) = im( f ) ( A, and picking some a ∈ A \ fix( f ), we have (1) (1) (1) (1) ∗(1) ca ∈/ PolA {fix( f )}, so ca ∈ OA \ { f } by Corollary 24.

(1) −1 Lemma 25. Let f ∈ OA be a retraction, I := x ∈ ﬁx( f ) Ax := f [{x}] \ {x} 6= ∅ . (1) ∗(1) ∗ Assume that M ⊆ OA is a transformation monoid with { f } ( M, then the centraliser M is ∗(1) ∗(1)∗(1) (1) a clone of idempotent operations, i.e., M = {idA}. So, if M = M , then M = OA .

Proof. According to Lemma7, we have to show that the functions in M∗ preserve all singletons of A. Thus, the goal is to show {t} ∈ [M•] ⊆ Inv Pol M• = Inv M∗ RA A A A for all t ∈ A, that is, to primitive positively define the singletons {t} from the graphs of (1) ∗(1) the functions in M. If t ∈ fix( f ), then ct ∈ { f } ⊆ M by Lemma5, and therefore ∗ (1) M ⊆ PolA{{t} | t ∈ fix( f )} by Lemma1 as im ct = {t}. Hence, it only remains to primitive positively define {t} ∈ [M•] for all t ∈ A \ fix( f ), and, in fact, it is sufficient for RA this to define {a} ∈ [M•] for some specific a ∈ A \ fix( f ). Namely, if {a} ∈ [M•] and RA RA ∗(1) we have any other t ∈ A \ (fix( f ) ∪ {a}), then there is ha,t ∈ { f } ⊆ M with ha,t(a) = t, so {t} = h [{a}] ∈ [M•] . Let j := f (a) so a ∈ A . If t ∈ A , too, then we define a,t RA j j ha,t ∈ PolA{fix( f )} by ha,t(a) := t and ha,t(x) := x for x ∈ A \ {a}. Then for all i ∈ I and Symmetry 2021, 13, 1471 17 of 40

x ∈ Ai we have ha,t(i) = i and the condition in Corollary 24 turns into ha,t(x) ∈ {i} ∪ Ai, ∗(1) which is satisfied for a as a, t ∈ Aj and clearly holds for all other x. So ha,t ∈ { f } . If t ∈ A` with ` ∈ I \ {j}, we put ha,t(j) := `, ha,t(x) := t for x ∈ Aj and ha,t(x) := x else. Clearly, ha,t ∈ PolA{fix( f )} as ha,t(j) = ` ∈ fix( f ) and ha,t(i) = i for all i ∈ I \ {j}. For such i and x ∈ Ai we have ha,t(x) = x ∈ {i} ∪ Ai, and for j and x ∈ Aj we have ∗(1) ha,t(x) = t ∈ {`} ∪ A`, so Corollary 24 yields that ha,t ∈ { f } . Now, to establish {a} ∈ [M•] for at least one a ∈ A \ fix( f ), we shall exploit the RA existence of h ∈ M \ { f }∗(1). According to Corollary 24 there may be two reasons why ∗(1) h ∈/ { f } . The first case is that h ∈/ PolA{fix( f )}. This means there is t ∈ fix( f ) such that (1) (1) ∗ a := h(t) ∈ A \ fix( f ). Then h ◦ ct = ca ∈ M, and Lemma1 shows M ⊆ PolA{a}, so (1) • {a} = im ca is primitive positively definable from M . ∗(1) The second case why h ∈/ { f } is that h ∈ PolA{fix( f )} but there is j ∈ I and a ∈ Aj with h(a) ∈/ f −1[{h(j)}]. Referring to Corollary 24, we distinguish as subcases whether −1 ` := h(j) ∈ I or not. First suppose that ` ∈ I, then h(a) ∈/ f [{`}] = {`} ∪ A`, that is, −1 −1 a ∈ h [B] 63 j where B := A \ (A` ∪ {`}). Hence, a ∈ h [B] ∩ {a, j} being a subset of −1 (1) A \ {j} ∩ {a, j} = {a}, i.e., {a} = h [B] ∩ {a, j}. We define g ∈ PolA {fix( f )} by g(a) := a and g(x) := j for x ∈ A \ {a}. For all i ∈ I and x ∈ Ai we have g(x) ∈ {a, j} ⊆ {j} ∪ Aj, which is in accordance with Corollary 24 since g(i) = j. Thus, g ∈ { f }∗(1) ⊆ M and im(g) = {a, j}. To define a function g˜ ∈ { f }∗(1) with im(g˜) = B, we observe that there is (1) −1 t ∈ fix( f ) \ {`}, for otherwise f = c` , contradicting h(a) ∈/ f [{h(j)}]. We let g˜(x) := x (1) for x ∈ fix( f ) \ {`} and g˜(`) := t ∈ fix( f ) \ {`}, then g˜ ∈ PolA {fix( f )}. For x ∈ A` we define g˜(x) := t ∈ {t} ∪ At, which agrees with Corollary 24 as g˜(`) = t. Finally, for x ∈ Ai where i ∈ I \ {`}, we put g˜(x) := x, which also complies with Corollary 24. Hence, g˜ ∈ { f }∗(1) ⊆ M and im(g˜) = B. Therefore, we get {a} = h−1[im(g˜)] ∩ im(g) ∈ [M•] . RA The remaining subcase of the second case is that h ∈ PolA{fix( f )} and ` ∈ fix( f ) \ I. By the condition in Corollary 24, we have h(a) ∈ A \ {h(j)} = A \ {`} for the violating −1 element a ∈ Aj with j ∈ I. Thus, we get a ∈ h [A \ {`}] ∩ {a, j} ⊆ A \ {j} ∩ {a, j} = {a}, that is, {a} = h−1[A \ {`}] ∩ {a, j}. To define a unary operation gˆ ∈ { f }∗(1) having im(gˆ) = A \ {`}, we note that I 6= ∅, as otherwise fix( f ) = A, i.e., f = idA and then (1) ∗(1) OA ⊇ M ) { f } would be violated. We pick some i ∈ I and define gˆ(`) := i and gˆ(x) := x for x ∈ A \ {`}. By Corollary 24 we have gˆ ∈ { f }∗(1) ⊆ M because ` ∈/ I. Clearly, im(gˆ) = A \ {`}, and so {a} = h−1[im(gˆ)] ∩ im(g) ∈ [M•] where we are re-using RA g ∈ { f }∗(1) ⊆ M with im(g) = {a, j} from above.

Lemma 25 covers in particular such retractions for which the image contains all elements except for one. For these we can show that they never witness maximal centralising monoids.

(1) Lemma 26. Let a ∈ A and f ∈ OA be a retraction with ﬁx( f ) = A \ {a}. Moreover, let (2) g ∈ OA be given by g(x, y) := f (x) if (x, y) 6= (a, a) and g(a, a) := a. Then g is a binary ∗(1) ∗(1) n (1)o idempotent operation and {g} = { f } ∪˙ ca .

Proof. Clearly, for every x ∈ A \ {a} we have g(x, x) = f (x) = x; moreover, g(a, a) = a, (1)∗ (1) ∗(1) so g ∈ CA is idempotent (Corollary7) and so CA ⊆ {g} . Consider now any h ∈ { f }∗(1); we have to show that h ∈ {g}∗. For this take (x, y) ∈ A2. First we assume that (x, y) 6= (a, a). There is t ∈ {x, y} with t ∈ A \ {a} = fix( f ). Since by Corollary 24 ∗(1) h ∈ { f } ⊆ PolA{fix( f )}, we have h(t) ∈ fix( f ), i.e., h(t) 6= a, so (h(x), h(y)) 6= (a, a). Now the definition of g implies g(h(x), h(y)) = f (h(x)) = h( f (x)) = h(g(x, y)). The remaining possibility is that (x, y) = (a, a), and here h(g(a, a)) = h(a) = g(h(a), h(a)), for g is idempotent. Therefore, h ∈ {g}∗. Symmetry 2021, 13, 1471 18 of 40

∗(1) (1) ∗ Conversely, let h ∈ {g} . If h = ct for some t 6= a, then h ∈ { f } because ∗(1) t belongs to A \ {a} = ﬁx( f ), cf. Lemma5. Now consider h ∈ {g} \ CA and any x ∈ A. As h is not constant, there is y ∈ A such that h(y) 6= h(x). It certainly follows that {(x, y), (h(x), h(y))} ⊆ A2\{(a, a)}, so h( f (x)) = h(g(x, y)) = g(h(x), h(y)) = f (h(x)). We conclude that h ∈ { f }∗(1).

(1) ∗(1) Corollary 27. Let f ∈ OA \ {idA} be any non-identical retraction, then { f } is maximal in the lattice of all centralising monoids on A witnessed by unary operations. In particular this holds (1) for retractions f ∈ OA with ﬁx( f ) = A \ {a} for some a ∈ A; however, these do not witness a maximal centralising monoid.

Proof. Maximality with respect to the closure system of the operator X 7→ X∗(1)∗(1) follows from Lemma 25; for functions with a single non-ﬁxed point non-maximality in general is evident from Lemma 26.

Lemma 25 and Corollary 27 deal with all non-identical retractions on any set and successfully explain our computational results regarding those centralising monoids on A = {0, 1, 2, 3} that can be witnessed by just unary operations. For retractions ﬁxing all but one element, we have that the centralising monoid is never maximal; for constant retractions it is known that the centralising monoid is always maximal, see Corollary 29 below. For the other types of centralising monoids witnessed by non-constant retractions, we currently do not know more than what is stated in Corollary 27. On a four-element set there are two types of such retractions, both having a two-element image. Our computations from [27] and Section4 show that the corresponding monoids are not contained in any proper centralising monoid witnessed by a unary or (idempotent) binary or majority operation on A = {0, 1, 2, 3}, but at the moment we cannot exclude that there might be a monoid witnessed by a semiprojection, which is properly larger. We conclude the discussion of monoids witnessed by retractions by giving a short and new proof of the fact that on sets with at least three elements every centralising monoid witnessed by a constant is actually maximal. This has been shown before in [21] (Theorem 5.1, p. 157).

Lemma 28. Let A be a set with {0, 1, 2} ⊆ A and M ⊆ OA be a set of operations satisfying ∗(1) n (1)o (1) (1) ∗ M ) c0 = T0 = PolA {{0}}, then M = JA.

n o∗(1) n o (1) = (1){{ }} = (1) = ∈ O(1) ∈ ( ) Proof. By Lemma5, c0 PolA 0 T0 f A 0 fix f , and (1) (1) (1) since there is s ∈ M \ T0 , there is s ∈ M with 0 ∈/ fix(s). Moreover, c0 ∈ T0 ⊆ M, (1) and a ∈/ fix c0 for any a ∈ A \ {0}. Hence, the initial precondition of Corollary 13 has (1) been established. We shall use the operations from T0 ⊆ M to obtain conditions (B’) (1) and (D’). For this let x, y, z, u ∈ A be pairwise distinct. If z = 0, then s0 ∈ T0 defined (1) by s0(0) := 0 and s0(a) := 1 for a ∈ A \ {0} shows (B’); if z 6= 0, then s1 ∈ T0 defined by s1(x) = s1(y) = s1(0) := 0 and s1(a) := 1 for a ∈ A \ {0, x, y} shows (B’). Similarly, if (1) (1) u = 0, then s0 ∈ T0 establishes (D’), and if u 6= 0 ,then s3 ∈ T0 given as s3(u) := 1 and ∗ s3(a) := 0 for a ∈ A \ {u} shows (D’). Now Corollary 13 entails M = JA.

Corollary 29 ([21] (Theorem 5.1, p. 157)). For any set A with |A| ≥ 3 and a ∈ A, the monoid (1) n (1)o∗(1) Ta = ca is a maximal centralising monoid.

(1) (1) ∗ Proof. If M ⊆ OA is a centralising monoid with M ) Ta , then we have M = JA from ∗∗(1) (1) Lemma 28, and so M = M = OA by a combination of Lemmas2 and9. Symmetry 2021, 13, 1471 19 of 40

3.4. Results We summarise the results of the current section in the following theorem.

(1) Theorem 30. Let A be a finite set and f ∈ OA \ {idA} be a unary minimal function from Theorem4. If • |A| ≥ 4 and f ∈ Sym A is a transposition of just two elements, • |A| ≥ 4 is even and f ∈ Sym A is a product of disjoint transpositions without fixed points, • f = f ◦ f is any (non-identical) retraction, ∗(1) (1) then { f } ( OA is a centralising monoid that is maximal among those monoids on A witnessed by unary operations. If |A| ≥ 3 and f is constant, then { f }∗(1) is always a maximal centralising monoid. Moreover, if • |A| ≥ 4 and f ∈ Sym A is a transposition of just two elements, • |A| ≥ 2 and f ∈ Sym A has a single fixed point, • |A| = 4 and f ∈ Sym A is a product of two disjoint transpositions (without fixed points), • f is a retraction with im( f ) = A \ {a} for some a ∈ A, then { f }∗(1) is not a co-atom (maximal) in the lattice of all centralising monoids on A.

Proof. The positive statements follow from Corollaries 22, 27 and 29; the negative statements from Lemma 16 and again Corollaries 22 and 27.

Corollary 31. On a four-element set no maximal centralising monoid is witnessed by a permutation nor a retraction with a three-element image; however all constants witness maximal centralising monoids. Moreover, every permutation of order 2 and every non-identical retraction witnesses a centralising monoid forming a co-atom in the lattice of all centralising monoids witnessed by unary operations.

4. Monoids Witnessed by Binary Idempotent Operations ∗(2) Each binary idempotent operation g ∈ CA satisﬁes g(x, x) = x for all x ∈ A. 2 Therefore, on A = {0, 1, 2, 3} there are 44 −4 = 412 = 16,777,216 binary idempotent operations, which are candidates for binary minimal functions. In a very similar way as shown in Listing1 we used a (slightly more complicated) c++ implementation to brute force enumerate all these binary operations g, and for each of them produced a characteristic 256 ∗(1) ∗(2) ∗(1) ∗(1) vector in {0, 1} of {g} . Whenever we found some g˜ ∈ CA with {g˜} = {g} for a previously enumerated g, we did not store {g˜}∗(1), that is, we dropped duplicate ∗(2) (1) rows from the context K2 := CA , OA , ⊥ . We also removed the binary projections that (1) 0 (1) describe the full transformation monoid OA . The result is a subcontext K2 = G, OA , ⊥ ∗(2) with a 10,263-element subset G ⊆ CA of binary idempotent non-projections that has the same closure system of monoids (intents) as K2. Using again algorithms from [35,36], we calculated that there are 17,013 centralising monoids on a four-element set that can be witnessed by some set of binary idempotent operations (and clearly 10,263 of them can be witnessed by just a single such operation). These numbers are of interest as they give the highest currently available lower bounds for the cardinality of the lattice CA of centraliser clones on A = {0, 1, 2, 3}, which is totally unexplored at the moment. We see a steep increase of these cardinalities as the size of A grows:

|A| : 2 3 4 [8] (p. 155), [3] (Figures 5 and 6, pp. 3158, 3162) [8] (p. 155), [9] (p. 125 et seqq.) |CA| : 25 2986 ≥ 17,013

Removing duplicate rows from K2 is just one way of simplifying a formal context, known as object clariﬁcation in the terminology of [35]. Another idea is to omit a row Symmetry 2021, 13, 1471 20 of 40

∗(2) ∗(1) T ∗(1) ∗(1) described by g ∈ CA if there is H ⊆ G \ {g} such that {g} = h∈H{h} = H , i.e., if the row can be obtained as an intersection of other rows. This operation is known as object or row reduction, see [35] (p. 24), and leaves only functions g ∈ G where the ∗(1) T monoid {g} is -irreducible in the lattice of centralising monoids belonging to K2. One 0 † ˜ (1) ˜ † can object reduce K2 to a context K2 = G, OA , ⊥ where G = 1236 and K2 has the 0 12 same closure system of monoids as K2 and K2. That is, only 1236 out of all 4 binary idempotent operations, those in G˜, sufﬁce to witness all 17,013 centralising monoids that can be witnessed by any selection of binary idempotent operations on {0, 1, 2, 3}. We have † listed these in the AppendixB; it is a simple task to reproduce K2 from them. 0 † In the same way as explained in Section 3.1 one can iterate over the rows of K2 (or K2) to ﬁnd the proper centralising monoids in the closure system that are maximal under set inclusion. We found that there are 612 centralising monoids that are inclusion maximal among those witnessed by binary idempotent operations, and their single witnesses are those operations in AppendixB that are marked by ↑. Again, these 612 monoids are not necessarily maximal centralising monoids, just candidates, they still need to be compared to the candidates witnessed by other types of minimal functions (unary, Mal’cev, majority— see Tables 4 and 5 in [27], and semiprojections). As the case of semiprojections has not been computed yet, we defer any comparisons of the different candidates for maximal centralising monoids to a point when all the data will be available. Let us summarise the main results of this section:

Theorem 32. On A = {0, 1, 2, 3} there are exactly 17,013 centralising monoids that can be witnessed by sets of binary idempotent operations; 612 of these are co-atoms under inclusion. All ˜ ∗(2) these monoids are witnessed by subsets of the 1236-element subset G ⊆ CA (see AppendixB, the 612 co-atoms are marked there by ↑ and correspond to the functions with indices 1, 4, 6, 10, 12, 19, 20, 21, 31, 33, 35, 36, 39, 44, 49, 50, 68, 69, 70, 71, 97, 135, 151, 153, 156, 159, 166, 168, 169, 347, 348, 351, 372, 377, 396, 412, 442, 454, 563, 618, 627, 657, 658, 659, 697, 762, 771, 774, 779, 780, 933, 934, 979, 981, 982, 983, 984, 986, 997, 1002, 1003, 1011, 1013, 1018, 1020, 1022, 1026, 1028, 1029, 1033, 1038, 1042, 1049, 1137, 1180, 1181, 1187 and their conjugates (cf. below and AppendixC) 11, 15, 16, 17, 22, 25, 26, 30, 42, 43, 48, 51, 53, 55, 56, 58, 61, 63, 65, 67, 75, 78, 81, 86, 87, 88, 92, 96, 98, 105, 107, 108, 110, 111, 113, 114, 118, 123, 124, 125, 126, 127, 128, 129, 130, 133, 136, 139, 143, 146, 148, 149, 150, 154, 157, 164, 165, 172, 173, 174, 177, 178, 180, 182, 183, 184, 186, 187, 188, 190, 191, 192, 196, 197, 198, 199, 200, 202, 203, 204, 208, 211, 214, 216, 217, 219, 222, 229, 230, 236, 237, 238, 240, 241, 242, 244, 245, 246, 249, 250, 251, 252, 253, 255, 256, 259, 260, 263, 264, 269, 270, 272, 276, 277, 279, 280, 281, 283, 284, 287, 288, 294, 297, 298, 299, 301, 303, 304, 306, 307, 308, 309, 313, 314, 320, 321, 322, 325, 326, 330, 333, 334, 335, 336, 337, 338, 339, 342, 343, 344, 345, 353, 354, 361, 362, 365, 366, 368, 369, 373, 376, 378, 382, 384, 385, 387, 389, 393, 395, 397, 399, 400, 401, 404, 405, 408, 410, 411, 413, 415, 416, 420, 421, 423, 426, 429, 439, 443, 446, 448, 449, 451, 452, 456, 459, 461, 462, 463, 466, 470, 471, 472, 475, 476, 477, 479, 480, 481, 483, 484, 485, 487, 502, 506, 510, 511, 512, 519, 520, 523, 527, 528, 529, 535, 536, 537, 539, 543, 544, 545, 546, 549, 550, 551, 554, 555, 557, 560, 561, 562, 569, 571, 572, 575, 577, 579, 580, 581, 582, 586, 595, 600, 601, 602, 604, 607, 608, 612, 613, 614, 615, 619, 620, 623, 625, 626, 629, 634, 635, 636, 639, 640, 642, 645, 646, 647, 648, 650, 655, 656, 660, 662, 669, 670, 671, 672, 673, 676, 678, 679, 682, 683, 684, 685, 686, 691, 696, 699, 700, 702, 703, 704, 707, 708, 709, 712, 714, 715, 716, 718, 722, 723, 724, 727, 728, 730, 731, 732, 733, 734, 735, 738, 739, 740, 741, 742, 744, 745, 748, 751, 752, 753, 754, 757, 759, 761, 764, 765, 766, 768, 775, 776, 777, 778, 782, 785, 786, 790, 791, 793, 794, 796, 800, 803, 804, 811, 812, 813, 815, 818, 819, 820, 822, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 846, 847, 848, 849, 851, 852, 854, 855, 858, 861, 865, 867, 868, 872, 873, 874, 875, 876, 879, 881, 882, 886, 887, 888, 889, 890, 891, 893, 894, 895, 896, 901, 913, 914, 915, 916, 917, 918, 922, 923, 925, 927, 939, 941, 943, 945, 946, 948, 949, 951, 952, 954, 955, 956, 962, 964, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 990, 991, 992, 993, 994, 995, 998, 999, 1000, 1006, 1007, 1015, 1021, 1023, 1030, 1034, 1037, 1043, 1044, 1045, 1046, 1056, 1059, 1060, 1062, 1067, 1069, 1070, 1071, 1072, 1078, 1080, 1081, 1082, 1084, 1087, 1088, 1089, 1095, 1097, 1099, 1100, 1106, 1107, 1108, 1109, 1110, 1111, 1118, Symmetry 2021, 13, 1471 21 of 40

1119, 1121, 1122, 1123, 1126, 1127, 1130, 1131, 1141, 1143, 1145, 1146, 1147, 1156, 1158, 1159, 1160, 1161, 1162, 1168, 1170, 1171, 1172, 1173, 1174, 1185, 1186, 1202, 1204, 1205, 1206, 1219, 1220, 1221, 1222, 1223, 1224, 1225, 1226, 1233, 1235, 1236).

Proof. Straightforward (time consuming) computation from the given data.

As in [27], we also use conjugation by inner automorphisms to simplify the situation. s (n) (n) We deﬁne the conjugate f ∈ OA of an n-ary operation f ∈ OA by a permutation s −1 n (n) s ∈ Sym A via f (x) := s( f (s ◦ x) for all x ∈ A . We say that operations f , g ∈ OA are conjugate if there is s ∈ Sym A such that g = f s, and we can hence form conjugacy classes of functions from this equivalence relation. It is helpful to extend the conjugation notation to sets F ⊆ OA by an element-wise deﬁnition: for s ∈ Sym A we stipulate Fs := { f s | f ∈ F}. Clearly, this conjugation notion respects any composition structure that F may have. The following observation is an immediate consequence from the fact D E D E ( ) −→ ( s) 7→ s that s : A; f f ∈OA A; f f ∈OA is an isomorphism and f f is injective.

Lemma 33 (cf. Observation 5.4 in [27]). For all n ∈ N, F ⊆ OA and permutations s ∈ Sym A s we have Fs∗(n) = F∗(n) .

s s∗(n) s ∗(n) T s∗(n) T ∗(n) Proof. We have F = { f | f ∈ F} = f ∈F{ f } = f ∈F { f } from [27]. By s s s hT ∗(n)i h ∗(n)i the injectivity of f 7→ f the latter equals f ∈F { f } = F .

Mainly relevant in the context of centralising monoids is the case n = 1 of Lemma 33, where it states that the conjugate by s of a centralising monoids witnessed by F is the centralising monoid witnessed by the conjugate Fs of F. As the centralising monoid F∗(1) is isomorphic to its conjugate via f 7→ f s, it is a useful initial step to simplify a large list of monoids given by witnesses via separating conjugacy classes of the witnesses. In a second turn one can then classify the remaining monoids according to element-wise conjugacy. In this respect we obtained the following results: the 10,263 idempotent binary non- 0 projections constituting the objects G of K2 can be classiﬁed into 2274 conjugacy types by inner automorphisms. Similarly, the 1236 binary operations in G˜ ⊆ G can be grouped together into 142 conjugacy classes (their representatives are marked in AppendixB by over- or underlining), and the 612 idempotent binary operations describing maximal elements with respect to K2 fall into 77 conjugacy types (representatives are marked in AppendixB by ↑ and over- or underlining, the monoids are listed in AppendixA, the classiﬁcation of witnesses into conjugacy classes is given in AppendixC). The respective numbers of monoids up to element-wise conjugacy are therefore at most as big as these values, but generally reduce further. For example, the 1236 monoids {g}∗(1) for g ∈ G˜ can be separated into 83 classes up to element-wise conjugacy. They correspond to those operations in AppendixB that are overlined. The other numbers are contained in the following theorem.

Theorem 34. Let A = {0, 1, 2, 3}. Up to element-wise conjugacy there are exactly 923 centralising ∗(2) monoids that can be witnessed by subsets of CA , 563 proper ones that can be witnessed by singletons, and 42 that are inclusion maximal proper monoids (among monoids witnessed by binary idempotent non-projections). These 42 monoids are shown in Appendix A.1 and correspond to those operations in AppendixB that are overlined and marked with ↑.

5. Monoids Witnessed by Mal’cev Operations of Boolean Groups In this section, we are going to consider a set A carrying a Boolean group hA; +, −, 0i, i.e., fulfilling the law x + x ≈ 0. Such a group is necessarily Abelian, and since every non- unit element has order two, there is a natural Z2-vector space structure on A by defining 1 · a := a and 0 · a := 0 for all a ∈ A. It follows from this that if A is finite, then |A| = 2d for some d ∈ N. As the group is Boolean, it satisfies −x ≈ x and any linear combinations Symmetry 2021, 13, 1471 22 of 40

simply reduce to sums. All identities holding in A clearly also transfer to its powers, and notationally we shall not distinguish between the (infix) addition on A or on a power, as it is common in linear algebra. Following case (III) of Rosenberg’s Theorem4, we consider the Mal’cev (minority) (3) operation f ∈ OA given as f (x, y, z) := x − y + z = x + y + z for x, y, z ∈ A. Our goal is to prove via Corollary 12 that { f }∗(1) always is a maximal centralising monoid on A, given |A| ≥ 4. From the theory of maximal clones on finite sets, the centraliser of the Mal’cev operation of any Abelian p-group for prime p is very well known, see, e.g., [38] (4.3.13 Lemma, p. 107) and note that the quaternary relation $G given in [38] (4.3.12 Definition, p. 106) is a variable permutation of the graph f • of the Mal’cev operation f of G. We here state the characterisation only for p = 2, where we can replace ‘−’ by ‘+’.

(3) Lemma 35. Let hA; +, −, 0i be a Boolean group and f ∈ OA be its Mal’cev operation and (n) ∗ g ∈ OA with n ∈ N. For L := { f } we have

g ∈ L ⇐⇒ ∀x, y, z ∈ An : g(x + y + z) = g( f ◦ (x, y, z)) = f (g(x), g(y), g(z)) = g(x) + g(y) + g(z) ⇐⇒ ∀x, y, z, u ∈ An : u = x + y + z =⇒ g(u) = g(x) + g(y) + g(z) ⇐⇒ ∀x, y, z, u ∈ An : x + y + z + u = 0 =⇒ g(x) + g(y) + g(z) + g(u) = 0 n ⇐⇒ g : hA; +, −, 0i −→ hA; +, −, 0i is Z2-afﬁne linear.

Since the centraliser L = { f }∗ of the Mal’cev operation of a p-group is the clone of affine (linear) functions, it will be necessary to review a few basic facts regarding affine functions and affine (in)dependence. Elements a1, ... , an in an affine space A are affinely independent if and only if for every linear combination 0 = λ1a1 + ··· + λnan with λ1 + ··· + λn = 0 necessarily all coefficients vanish: λ1 = ··· = λn = 0. Rephrasing this, they are affinely dependent precisely if there is a linear combination 0 = λ1a1 + ··· + λnan with λ1 + ··· + λn = 0 with some non-zero coefficients in it. Of course, we can remove all the coefficients which are zero from the sum without changing this condition. Thus, {a1,..., an} are affinely dependent if and only if there is a subset ∅ 6= S ⊆ {a1,..., an} and non-zero coefficients λs for s ∈ S such that 0 = ∑s∈S λss and ∑s∈S λs = 0. In the context of an affine space over Z2, all λs ∈ Z2 \ {0} = {1} are equal to 1. Hence, over Z2 a set {a1,..., an} is affinely dependent if there is a non-empty subset S ⊆ {a1,..., an} with ∑s∈S s = 0 and 0 = ∑s∈S 1 = |S| mod 2. Thus, a1, ... , an are affinely independent over Z2 if every non-empty sum of evenly many of these elements is distinct from the zero-vector 0. The following special cases of affine independence over Z2 will be relevant for our proof: a1, a2 ∈ A are affinely independent if and only if a1 6= a2 since a1 + a2 = 0 is equivalent to a1 = a2; three general elements a1, a2, a3 ∈ A are affinely independent if and only if |{a1, a2, a3}| = 3, i.e., they are pairwise distinct. Moreover, a1, a2, a3, a4 ∈ A are affinely independent exactly if |{a1, a2, a3, a4}| = 4 and a1 + a2 + a3 + a4 6= 0. Therefore, four pairwise distinct elements a1, a2, a3, a4 of A are affinely independent precisely if a1 + a2 + a3 + a4 6= 0. The importance of affine independence in our situation comes from the following extendability property, which is part of a typical introductory course to linear algebra (see, e.g., Satz 6.3.6 in [39] (p. 156)).

Proposition 36. Let A and A0 be affine spaces over the same field and B ⊆ A be an affine basis 0B of A. Then for every partial assignment (g(b))b∈B ∈ A there is a unique extension to an affine linear map g : A −→ A0. Since every affinely independent set I ⊆ A can be extended to an affine basis B ⊆ A, every 0 I 0 partial assignment (g(x))x∈I ∈ A is extendable to an affine map g : A −→ A . Symmetry 2021, 13, 1471 23 of 40

Lemma 37. For a Boolean group hA; +, −, 0i with |A| ≥ 4 and its associated Mal’cev operation (3) (1) ∗(1) (1) ∗ f ∈ OA , let M ⊆ OA be a monoid satisfying M ) { f } = L ; then M = JA.

(1) ∗ Proof. By the assumption there is some unary h ∈ M ⊆ OA with h ∈/ { f } , i.e., the condition given in Lemma 35 is falsiﬁed by h. We proceed through a series of claims.

Claim 1. There are pairwise distinct afﬁnely dependent x, y, z, u ∈ A for which there is a function g ∈ M with (x, y) ∈ ker(g) and (z, u) ∈/ ker(g).

Proof. To prove this claim we use the operation h ∈/ { f }∗(1). This means there are (affinely dependent) x + y + z + u = 0 in A for which h(x) + h(y) + h(z) + h(u) 6= 0. If any two elements from (x, y, z, u) were equal, then because of x + y + z + u = 0 also the other two would have to be equal. It would follow that h(x) + h(y) + h(z) + h(u) = 0 for it would be a sum of two pairs of equal summands. This contradicts the assumption on x, y, z, u. Therefore, we know that x + y + z + u = 0 and h(x) + h(y) + h(z) + h(u) 6= 0 and x, y, z, u are pairwise distinct. These conditions are completely symmetric under permutation of the four values. If there are any two that are sent by h to the same value, then because of 0 6= h(x) + h(y) + h(z) + h(u) the other two are not in the kernel of h, and h and some permutation of (x, y, z, u) will show Claim1. Now let us assume that all four h-values are pairwise distinct. Then h(x), h(y), h(z), h(u) are affinely independent, and by Proposition 36 we choose some affine linear map g ∈ L(1) ⊆ M satisfying g(h(y)) := h(x) and g(h(w)) := h(w) for w ∈ {x, z, u}. Then g ◦ h ∈ M and g ◦ h(x) = h(x) = g ◦ h(y), g ◦ h(z) = h(z) 6= h(u) = g ◦ h(u), i.e., (x, y) ∈ ker(g ◦ h) and (z, u) ∈/ ker(g ◦ h). So g ◦ h will show Claim1.

Claim 2. For all pairwise distinct afﬁnely dependent x, y, z, u ∈ A there is a function g ∈ M with (x, y) ∈ ker(g) and (z, u) ∈/ ker(g).

Proof. By Claim1 there are affinely dependent pairwise distinct elements x, y, z, u ∈ A and g ∈ M with g(x) = g(y) and g(z) 6= g(u). Now consider any other affinely dependent quadruple (a, b, c, d) with pairwise distinct entries. Dependency means precisely that x + y + z + u = 0 and a + b + c + d = 0, or equivalently, u = x + y + z and d = a + b + c. However, as a, b, c are pairwise distinct, these points are affinely independent. Via Proposition 36 we choose an affine linear map g0 ∈ L(1) ⊆ M such that g0(a) = x, g0(b) = y and g0(c) = z. Since g0 ∈ L(1) = { f }∗, it follows (see Lemma 35) that g0(d) = g0(a) + g0(b) + g0(c) = x + y + z = u. Thus, g ◦ g0(a) = g(x) = g(y) = g ◦ g0(b) and g ◦ g0(c) = g(z) 6= g(u) = g ◦ g0(d), wherefore we obtain (a, b) ∈ ker(g ◦ g0) but (c, d) ∈/ ker(g ◦ g0). Hence, g ◦ g0 ∈ M shows Claim2.

Claim 3. For all pairwise distinct elements x, y, z, u ∈ A there exists a function g ∈ M such that (x, y) ∈ ker(g) and (z, u) ∈/ ker(g); i.e., g(x) = g(y) but g(z) 6= g(u).

Proof. Let x, y, z, u ∈ A be pairwise distinct. If they are afﬁnely dependent, the result follows from Claim2; hence suppose that x, y, z, u are afﬁnely independent. In this case we use Proposition 36 to construct g ∈ L(1) ⊆ M with g(x) = g(y) = g(z) = 0 6= g(u), so that (x, y) ∈ ker(g) and (z, u) ∈/ ker(g).

We now take advantage of Corollary 12 to complete the proof of Lemma 37. Condi- tion (D’) is provided by Claim3; condition (B’) holds as any pairwise distinct x, y, z ∈ A are affinely independent, and so we may again rely on Proposition 36 to pick some g ∈ L(1) ⊆ M with g(y) = g(x) 6= g(z). Similarly, for each a ∈ A this single point is affinely independent, and so Proposition 36 can be invoked to define some g ∈ L(1) ⊆ M ∗ with g(a) 6= a, i.e., a ∈/ fix(g). Consequently, Corollary 12 implies M = JA. Symmetry 2021, 13, 1471 24 of 40

(3) Theorem 38. Let hA; +, −, 0i be a Boolean group with |A| ≥ 4 and f ∈ OA be the associated Mal’cev operation. Then L(1) = { f }∗(1) is a maximal centralising monoid on A.

(1) (1) ∗ Proof. If M ⊆ OA is a centralising monoid satisfying L ( M, then M = JA by (1) Lemma 37. We now use Lemmas2 and9 to infer M = OA . Moreover, |A| ≥ 4 > 2 (1) (1) ensures that L ( OA . Namely, take pairwise distinct x, y, z ∈ A and let u := x + y + z. Since |{x, y, z}| = 3, u ∈/ {x, y, z}. We define h(u) 6= 0 and h(a) := 0 for a ∈ A \ {u}, in particular h(x) = h(y) = h(z) = 0. If h were affine linear, then, by Lemma 35, we would have h(u) = h(x) + h(y) + h(z) = 0 + 0 + 0 = 0, contradicting the definition of h. Hence, (1) (1) h ∈ OA \ L .

Remark 39. Four-element sets are the smallest ones where Theorem 38 (non-vacuously) holds. Three-element sets do not support Boolean groups—for this reason monoids witnessed by Mal’cev (1) (1) operations did not appear in the analyses given in [21,23]; and for |A| ≤ 2 we have L = OA , the full transformation monoid. Hence, in these cases { f }∗(1) = L(1) fails to be a maximal centralising monoid.

Supplementary Materials: The following are available online at https://www.mdpi.com/article/10 .3390/sym13081471/s1. Author Contributions: Conceptualisation, software, data curation, and writing—first draft, M.B.; project administration, validation, and visualisation, E.V.-G.; methodology, investigation, formal analysis, and writing—review and editing, M.B. and E.V.-G. Both authors have read and agreed to the published version of the manuscript. Funding: The research of M. Behrisch received partial funding support by BMBWF, through OeAD- GmbH project KONTAKT CZ 02/2019 ‘Function algebras and ordered structures related to logic and data fusion’. Open access funding was generously provided by TU Wien. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Additional data supporting and complementing our results is available as supplementary files. Acknowledgments: Both authors gladly acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Programme. Edith Vargas-García gratefully acknowledges support by the Asociación Mexicana de Cultura A.C. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Symmetry 2021, 13, 1471 25 of 40

Appendix A Appendix A.1

Centralising monoids being maximal w.r.t. K2, witnessed by 77 conjugacy types of binary idempotent functions gi (part 1: monoids representing one of 42 conjugacy classes). The indices i ∈ {1, . . . , 1236} of the witnesses refer to the overlined operations listed in AppendixB; the unary operations u in the monoids are coded via their base-4- 3 3−j representation ∑j=0 u(j)4 .

∗(1) {g1} = {0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 24, 27, 28, 30, 32, 33, 35, 36, 39, 44, 45, 48, 49, 50, 52, 54, 56, 57, 85, 170, 255} ∗(1) {g4} = {0, 3, 4, 7, 8, 11, 16, 19, 24, 27, 32, 35, 36, 39, 85, 170, 252, 255} ∗(1) {g6} = {0, 1, 2, 3, 5, 10, 11, 15, 16, 18, 19, 26, 27, 31, 32, 33, 37, 48, 49, 53, 85, 170, 171, 175, 186, 255} ∗(1) {g10} = {0, 3, 11, 16, 19, 27, 32, 85, 170, 171, 252, 254, 255} ∗(1) {g12} = {0, 4, 8, 11, 16, 24, 27, 32, 36, 85, 170, 239, 248, 251, 255} ∗(1) {g19} = {0, 1, 10, 11, 16, 26, 27, 85, 160, 161, 170, 171, 176, 177, 186, 255} ∗(1) {g20} = {0, 10, 11, 15, 16, 26, 27, 31, 85, 160, 170, 186, 240, 255} ∗(1) {g21} = {0, 2, 3, 8, 10, 11, 12, 14, 15, 16, 18, 19, 24, 26, 27, 28, 30, 31, 85, 160, 162, 163, 168, 170, 171, 172, 174, 175, 240, 242, 243, 248, 250, 251, 252, 254, 255} ∗(1) {g31} = {0, 1, 2, 3, 5, 7, 10, 11, 15, 17, 19, 27, 34, 35, 39, 51, 85, 87, 95, 119, 170, 171, 175, 187, 255} ∗(1) {g33} = {0, 1, 3, 8, 9, 11, 17, 19, 25, 27, 51, 59, 85, 87, 119, 162, 170, 255} ∗(1) {g35} = {0, 1, 3, 11, 17, 19, 27, 51, 85, 87, 119, 163, 170, 171, 187, 255} ∗(1) {g36} = {0, 8, 17, 19, 25, 27, 49, 51, 57, 59, 85, 93, 162, 170, 247, 255} ∗(1) {g39} = {0, 2, 3, 8, 10, 11, 12, 14, 15, 19, 27, 85, 87, 93, 95, 160, 162, 163, 168, 170, 171, 172, 174, 175, 240, 242, 243, 245, 247, 248, 250, 251, 252, 253, 254, 255} ∗(1) {g44} = {0, 4, 8, 12, 16, 24, 27, 32, 36, 39, 48, 85, 170, 223, 239, 247, 251, 255} ∗(1) {g49} = {0, 8, 16, 24, 27, 48, 85, 162, 170, 174, 223, 251, 255} ∗(1) {g50} = {0, 8, 12, 16, 19, 24, 27, 85, 162, 170, 174, 220, 223, 243, 251, 255} ∗(1) {g68} = {0, 1, 2, 3, 4, 8, 12, 21, 22, 23, 25, 27, 29, 30, 42, 63, 85, 86, 87, 89, 91, 93, 94, 106, 127, 170, 255} ∗(1) {g69} = {0, 4, 8, 12, 21, 23, 25, 27, 42, 61, 63, 85, 89, 106, 170, 255} ∗(1) {g70} = {0, 3, 4, 8, 23, 27, 85, 87, 89, 91, 170, 252, 253, 255} ∗(1) {g71} = {0, 1, 2, 3, 5, 6, 7, 10, 11, 15, 21, 22, 23, 26, 27, 31, 42, 43, 47, 63, 85, 86, 87, 90, 91, 95, 106, 107, 111, 127, 170, 171, 175, 191, 255} ∗(1) {g97} = {0, 21, 27, 42, 45, 54, 63, 85, 106, 170, 191, 213, 255} Symmetry 2021, 13, 1471 26 of 40

∗(1) {g151} = {0, 1, 2, 3, 6, 7, 11, 27, 84, 85, 168, 170, 252, 255} ∗(1) {g153} = {0, 5, 10, 11, 26, 27, 80, 85, 160, 161, 170, 175, 176, 177, 250, 255} ∗(1) {g156} = {0, 1, 2, 3, 4, 5, 8, 10, 11, 12, 14, 15, 26, 27, 30, 31, 80, 81, 84, 85, 160, 162, 163, 168, 170, 171, 172, 174, 175, 240, 242, 243, 248, 250, 251, 252, 254, 255} ∗(1) {g159} = {0, 1, 2, 3, 6, 11, 27, 84, 85, 87, 168, 170, 252, 253, 255} ∗(1) {g166} = {0, 4, 8, 9, 11, 12, 24, 25, 27, 56, 57, 59, 81, 85, 93, 162, 170, 243, 247, 255} ∗(1) {g168} = {0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 24, 27, 80, 81, 83, 84, 85, 87, 92, 93, 95, 160, 162, 163, 168, 170, 171, 172, 174, 175, 240, 241, 242, 243, 244, 245, 247, 248, 250, 251, 252, 253, 254, 255} ∗(1) {g169} = {0, 5, 10, 15, 27, 80, 85, 95, 160, 170, 175, 228, 240, 245, 250, 255} ∗(1) {g347} = {0, 5, 10, 11, 14, 15, 16, 21, 26, 27, 30, 31, 32, 37, 42, 48, 53, 63, 64, 69, 74, 75, 78, 79, 80, 85, 90, 91, 94, 95, 96, 101, 106, 112, 117, 127, 128, 133, 138, 144, 149, 154, 160, 165, 170, 192, 197, 207, 208, 213, 223, 240, 245, 255} ∗(1) {g348} = {0, 5, 10, 16, 21, 27, 32, 42, 64, 69, 78, 80, 85, 95, 117, 127, 128, 138, 160, 170, 175, 186, 191, 213, 223, 234, 239, 245, 250, 255} ∗(1) {g351} = {0, 5, 10, 15, 16, 21, 27, 30, 32, 42, 47, 48, 58, 63, 64, 69, 80, 85, 128, 138, 143, 160, 170, 175, 176, 186, 191, 192, 202, 207, 224, 234, 239, 240, 250, 255} ∗(1) {g372} = {0, 5, 11, 14, 16, 21, 27, 30, 64, 69, 75, 78, 80, 85, 91, 94, 170, 175, 186, 191, 234, 239, 250, 255} ∗(1) {g377} = {0, 21, 22, 27, 64, 85, 106, 149, 170, 191, 233, 234, 255} ∗(1) {g396} = {0, 21, 23, 27, 39, 42, 43, 61, 62, 63, 64, 85, 106, 128, 149, 170, 192, 255} ∗(1) {g412} = {0, 21, 22, 23, 25, 26, 27, 29, 30, 31, 37, 38, 39, 41, 42, 43, 45, 46, 47, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 85, 128, 170, 192, 255} ∗(1) {g454} = {0, 27, 45, 54, 85, 106, 127, 149, 170, 191, 213, 234, 255} ∗(1) {g563} = {0, 5, 10, 27, 78, 80, 85, 95, 160, 170, 175, 177, 228, 245, 250, 255} ∗(1) {g618} = {0, 4, 8, 19, 27, 85, 93, 116, 170, 205, 243, 251, 255} ∗(1) {g658} = {0, 20, 23, 24, 27, 36, 39, 40, 43, 65, 66, 77, 78, 85, 113, 114, 125, 126, 129, 130, 141, 142, 170, 177, 178, 189, 190, 212, 215, 216, 219, 228, 231, 232, 235, 255} ∗(1) {g697} = {0, 5, 27, 30, 75, 78, 80, 85, 170, 175, 177, 180, 225, 228, 250, 255} ∗(1) {g762} = {0, 10, 17, 27, 68, 78, 85, 95, 160, 170, 177, 187, 228, 238, 245, 255} ∗(1) {g1187} = {0, 27, 45, 54, 78, 85, 99, 120, 135, 156, 170, 177, 201, 210, 228, 255} Symmetry 2021, 13, 1471 27 of 40

Appendix A.2 ∗(1) Centralising monoids Mi := {gi} being maximal w.r.t. K2, witnessed by 77 conjugacy types of binary idempotent functions gi (part 2: monoids conjugate to the 42 ones shown in Appendix A.1). The indices i ∈ {1, . . . , 1236} of the witnesses refer to the (over- or underlined) operations listed in AppendixB; the unary operations u in the monoids are 3 3−j coded via their base-4-representation ∑j=0 u(j)4 .

(321) M135 = {0, 16, 21, 27, 31, 32, 42, 46, 48, 59, 63, 85, 170, 213, 223, 255} = M69 (12) M442 = {0, 27, 41, 42, 85, 106, 127, 128, 149, 170, 213, 214, 255} = M377 (0231) M627 = {0, 4, 8, 27, 85, 89, 93, 120, 170, 201, 243, 251, 255} = M97 M657 = {0, 3, 7, 11, 19, 23, 27, 35, 39, 43, 67, 71, 75, 83, 85, 87, 91, 99, 103, 107, 131, (0321) 135, 139, 147, 151, 155, 163, 167, 170, 171, 252, 253, 254, 255} = M412 (03) M659 = {0, 3, 23, 27, 39, 43, 67, 85, 86, 87, 131, 169, 170, 171, 252, 253, 254, 255} = M396 (310) M771 = {0, 15, 16, 27, 31, 80, 85, 90, 95, 154, 158, 170, 207, 219, 223, 255} = M35 (310) M774 = {0, 11, 15, 16, 27, 31, 80, 85, 91, 95, 170, 174, 203, 207, 219, 223, 251, 255} = M33 M779 = {0, 10, 11, 14, 15, 26, 27, 30, 31, 74, 75, 78, 79, 85, 90, 91, 94, 95, 160, 161, 164, (123) 165, 170, 176, 177, 180, 181, 224, 225, 228, 229, 240, 241, 244, 245, 255} = M658 M780 = {0, 2, 3, 8, 10, 11, 12, 14, 15, 26, 27, 30, 31, 74, 75, 78, 79, 85, 86, 87, 89, 90, 91, 93, 94, 95, 160, 162, 163, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 240, (02)(13) 242, 243, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255} = M347 (0123) M933 = {0, 23, 27, 63, 85, 89, 113, 125, 149, 170, 177, 189, 215, 219, 243, 255} = M19 (0123) M934 = {0, 23, 27, 60, 85, 105, 113, 125, 150, 170, 177, 189, 195, 215, 219, 255} = M153 (0231) M979 = {0, 27, 42, 43, 85, 87, 103, 155, 168, 170, 171, 213, 232, 234, 255} = M12 (02)(13) M981 = {0, 26, 27, 32, 85, 101, 138, 154, 170, 186, 202, 218, 239, 255} = M151 M982 = {0, 26, 27, 42, 43, 85, 101, 152, 154, 155, 168, 170, 171, 216, (0231) 218, 232, 234, 255} = M4 (02)(13) M983 = {0, 26, 27, 32, 85, 101, 117, 138, 154, 170, 186, 202, 223, 239, 255} = M159 (0231) M984 = {0, 26, 27, 42, 85, 101, 117, 154, 155, 170, 223, 234, 255} = M10 (013) M986 = {0, 27, 32, 85, 117, 138, 159, 170, 186, 202, 223, 239, 255} = M377 Symmetry 2021, 13, 1471 28 of 40

(0132) M997 = {0, 21, 27, 59, 63, 85, 89, 166, 170, 174, 213, 219, 251, 255} = M70 (310) M1002 = {0, 21, 27, 31, 53, 59, 63, 85, 89, 93, 166, 170, 174, 247, 251, 255} = M50 (310) M1003 = {0, 21, 27, 59, 63, 85, 89, 127, 166, 170, 174, 251, 255} = M49 M1011 = {0, 2, 3, 27, 32, 34, 35, 48, 50, 51, 85, 87, 117, 119, 136, 138, 139, 147, 168, 170, 171, (320) 184, 186, 187, 204, 206, 207, 221, 223, 236, 238, 239, 252, 253, 254, 255} = M351 M1013 = {0, 2, 25, 27, 32, 34, 57, 59, 85, 87, 117, 119, 136, 138, 145, 147, 168, 170, 177, (0132) 179, 221, 223, 253, 255} = M372 (210) M1018 = {0, 25, 27, 42, 59, 85, 102, 153, 155, 170, 187, 213, 238, 255} = M20 M1020 = {0, 24, 26, 27, 32, 48, 85, 101, 138, 152, 154, 155, 170, 186, 207, 216, 218, 219, (210) 239, 255} = M166 (210) M1022 = {0, 24, 27, 40, 43, 48, 85, 101, 154, 170, 207, 216, 219, 232, 235, 255} = M36 (310) M1026 = {0, 2, 3, 27, 35, 85, 86, 137, 168, 169, 170, 252, 255} = M377 M1028 = {0, 2, 3, 27, 32, 34, 48, 51, 85, 86, 87, 101, 102, 117, 119, 136, 138, 153, 154, 168, (013) 169, 170, 177, 204, 207, 221, 223, 252, 253, 255} = M348 M1029 = {0, 2, 3, 26, 27, 32, 34, 35, 48, 50, 51, 85, 86, 87, 101, 102, 103, 117, 118, 119, 136, 138, 139, 153, 154, 155, 168, 169, 170, 171, 184, 185, 186, 187, 204, 206, (210) 207, 221, 222, 223, 236, 237, 238, 239, 252, 253, 254, 255} = M168 M1033 = {0, 2, 25, 27, 32, 34, 57, 59, 85, 86, 87, 101, 102, 103, 117, 118, 119, 136, 138, 153, 154, 155, 168, 169, 170, 171, 185, 186, 187, 221, 222, 223, 237, 238, 239, (210) 253, 254, 255} = M156 M1038 = {0, 3, 27, 43, 48, 51, 85, 86, 87, 101, 102, 103, 117, 118, 119, 153, 154, 155, 169, 170, 171, 185, 186, 187, 204, 207, 221, 222, 223, 237, 238, 239, (210) 252, 253, 254, 255} = M39 M1042 = {0, 25, 26, 27, 41, 42, 43, 57, 58, 59, 85, 86, 87, 101, 102, 103, 117, 118, 119, 153, 154, 155, 169, 170, 171, 185, 186, 187, 221, 222, 223, 237, 238, 239, (210) 253, 254, 255} = M21 (01) M1049 = {0, 27, 32, 48, 85, 138, 156, 170, 186, 207, 210, 239, 255} = M454 (321) M1137 = {0, 27, 39, 60, 78, 85, 105, 114, 141, 150, 170, 177, 195, 216, 228, 255} = M697 (13) M1180 = {0, 27, 40, 60, 78, 85, 105, 125, 130, 150, 170, 177, 195, 215, 228, 255} = M563 (02) M1181 = {0, 27, 40, 60, 78, 85, 105, 125, 130, 150, 170, 190, 195, 215, 235, 255} = M169 Symmetry 2021, 13, 1471 29 of 40

Appendix B ˜ † Value tables of 1236 binary idempotent operations from G of K2: 1↑: 2: 3: 4↑: 5: 6↑: 7: 8: 9: 10↑: 11↑: 12↑: 13: 14: 15↑: 16↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0020 0020 0020 0020 0021 0022 0022 0022 0022 0022 0023 0023 0320 0323 2222 2222 0003 0013 0203 3333 0013 0023 0033 0223 2223 3333 0033 3333 0003 0033 0003 0023 17↑: 18: 19↑: 20↑: 21↑: 22↑: 23: 24: 25↑: 26↑: 27: 28: 29: 30↑: 31↑: 32: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0100 0100 0100 0100 0100 0100 0100 0100 0101 0101 0101 0101 0101 0101 0101 2222 2222 2222 2222 2222 2223 3320 3323 3323 0020 0020 0020 0020 0020 0022 0022 0033 2203 2223 3323 3333 3333 3333 0033 3333 0103 0113 0303 1113 3333 0123 3333 33↑: 34: 35↑: 36↑: 37: 38: 39↑: 40: 41: 42↑: 43↑: 44↑: 45: 46: 47: 48↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0101 0101 0101 0101 0101 0101 0102 0102 0103 0103 0103 0103 0103 0103 0103 2222 2222 2222 2222 2222 2222 2222 0020 2222 0020 0020 0023 0320 0323 0323 2222 0103 0113 0123 0303 1113 2323 3333 0203 3333 0303 3333 0333 0303 0333 3333 0303 49↑: 50↑: 51↑: 52: 53↑: 54: 55↑: 56↑: 57: 58↑: 59: 60: 61↑: 62: 63↑: 64: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0103 0103 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 2222 2222 0120 0120 0120 0121 0123 0123 0220 0220 1121 1121 2222 2222 2222 3223 0333 3333 0003 0013 3333 3333 0033 3333 0003 3333 0003 3333 0003 0033 3333 3333 65↑: 66: 67↑: 68↑: 69↑: 70↑: 71↑: 72: 73: 74: 75↑: 76: 77: 78↑: 79: 80: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0120 0120 0120 0121 0121 0121 0122 0122 0122 0122 0123 0123 0123 0222 0222 0222 0103 0113 3333 0113 0333 3333 0123 0133 0333 3333 0133 0333 3333 0113 0123 0133 81↑: 82: 83: 84: 85: 86↑: 87↑: 88↑: 89: 90: 91: 92↑: 93: 94: 95: 96↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0222 0222 0222 0222 0223 0323 2222 2222 2222 2222 2222 2222 2222 2222 2223 3323 0223 0323 0333 3333 0333 0333 0103 0113 0123 0133 0333 2223 3323 3333 3333 3333 97↑: 98↑: 99: 100: 101: 102: 103: 104: 105↑: 106: 107↑: 108↑: 109: 110↑: 111↑: 112: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0113 0113 0113 0113 0113 0113 0113 0113 0120 0120 0120 0120 0120 0120 0120 0120 0122 0123 0123 0222 0223 0223 2222 2222 0220 0220 0220 0222 0222 0222 2222 2222 0323 0333 3333 0333 0333 3333 0333 3333 0003 0203 3333 0023 0223 3333 0003 0223 113↑: 114↑: 115: 116: 117: 118↑: 119: 120: 121: 122: 123↑: 124↑: 125↑: 126↑: 127↑: 128↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0120 0121 0121 0121 0121 0121 0121 0121 0121 0122 0122 0122 0122 0122 0122 0123 2222 0222 0222 0222 0222 0223 2222 2222 2222 0220 0222 0222 0222 2222 2222 0220 3333 0123 0323 0333 3333 0133 0123 0323 3333 3333 0223 0333 3333 2223 3333 0303 129↑: 130↑: 131: 132: 133↑: 134: 135↑: 136↑: 137: 138: 139↑: 140: 141: 142: 143↑: 144: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0130 0220 0222 0222 0222 0223 0223 0323 2222 2222 2222 2222 2222 2222 2223 3223 0320 3333 0323 0333 3333 0333 3333 0323 0303 0323 0333 2323 3323 3333 3333 3333 0003 145: 146↑: 147: 148↑: 149↑: 150↑: 151↑: 152: 153↑: 154↑: 155: 156↑: 157↑: 158: 159↑: 160: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0130 0133 0133 0133 0133 0133 1100 1100 1100 1100 1100 1100 1100 1101 1101 1101 2222 0323 2222 2222 2222 3323 2220 2221 2222 2222 2222 2222 3322 2022 2220 2221 3333 0333 0303 0333 3333 3333 3333 3333 2233 3303 3313 3333 3333 3333 3333 3333 161: 162: 163: 164↑: 165↑: 166↑: 167: 168↑: 169↑: 170: 171: 172↑: 173↑: 174↑: 175: 176: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 1101 1101 1101 1101 1101 1101 1101 1101 1110 1110 1110 1110 1110 1110 1110 2222 2222 2222 2222 2222 2222 2222 2222 2322 1120 1220 2020 2022 2220 2221 2222 1103 1303 2333 3003 3033 3303 3313 3333 3333 3333 3333 3333 3333 3333 3333 3033 Symmetry 2021, 13, 1471 30 of 40

177↑: 178↑: 179: 180↑: 181: 182↑: 183↑: 184↑: 185: 186↑: 187↑: 188↑: 189: 190↑: 191↑: 192↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 1110 1110 1110 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2222 2222 2222 2222 3222 0020 0020 0020 0020 0020 0020 0020 0021 0022 0022 0022 3233 3303 3313 3333 3333 0003 0103 0303 1013 1113 3133 3333 3333 0023 0033 0123 193: 194: 195: 196↑: 197↑: 198↑: 199↑: 200↑: 201: 202↑: 203↑: 204↑: 205: 206: 207: 208↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0022 0022 0022 0022 0023 0023 0023 0120 0120 0120 0121 0121 0122 0122 0122 0123 0223 1133 2223 3333 0033 0333 3333 0003 0113 3333 1113 3333 0123 0133 3333 0033 209: 210: 211↑: 212: 213: 214↑: 215: 216↑: 217↑: 218: 219↑: 220: 221: 222↑: 223: 224: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0123 0123 0123 0123 0123 0220 0220 0220 0222 0222 0222 0222 0222 0222 0222 0222 0133 0333 1133 3133 3333 0003 0303 3333 0023 0123 0223 0323 0333 1113 3133 3333 225: 226: 227: 228: 229↑: 230↑: 231: 232: 233: 234: 235: 236↑: 237↑: 238↑: 239: 240↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0223 0223 0320 0323 0323 0323 1020 1020 1021 1021 1120 1121 1121 1121 1121 1121 0333 3333 3333 0033 0333 3333 1113 3333 0003 3333 3333 0003 0113 0333 1003 1113 241↑: 242↑: 243: 244↑: 245↑: 246↑: 247: 248: 249↑: 250↑: 251↑: 252↑: 253↑: 254: 255↑: 256↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1121 1121 1122 1122 1122 1122 1122 1122 1122 1123 1123 1123 1221 1221 1221 1221 3113 3333 0033 0123 1123 1133 2123 2223 3333 1133 3133 3333 1333 2333 3003 3333 257: 258: 259↑: 260↑: 261: 262: 263↑: 264↑: 265: 266: 267: 268: 269↑: 270↑: 271: 272↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1222 1222 1222 1222 1222 1222 1222 1222 1321 1322 1323 1323 2020 2020 2020 2020 1223 1233 1313 1333 2333 3033 3313 3333 3333 1333 1333 3333 1313 3033 3233 3333 273: 274: 275: 276↑: 277↑: 278: 279↑: 280↑: 281↑: 282: 283↑: 284↑: 285: 286: 287↑: 288↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2022 2022 2022 2022 2022 2022 2022 2022 2121 2121 2121 2122 2122 2122 2122 2122 1333 2023 2033 3003 3033 3233 3303 3333 1113 3113 3333 0003 0123 0333 1123 2123 289: 290: 291: 292: 293: 294↑: 295: 296: 297↑: 298↑: 299↑: 300: 301↑: 302: 303↑: 304↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2122 2122 2122 2123 2123 2220 2220 2220 2220 2220 2221 2221 2221 2221 2221 2222 3123 3133 3333 3133 3333 3033 3233 3303 3313 3333 1333 2333 3303 3313 3333 0003 305: 306↑: 307↑: 308↑: 309↑: 310: 311: 312: 313↑: 314↑: 315: 316: 317: 318: 319: 320↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 0013 0023 0033 0103 0113 0123 0133 0203 0223 0303 0323 0333 1003 1013 1103 1113 321↑: 322↑: 323: 324: 325↑: 326↑: 327: 328: 329: 330↑: 331: 332: 333↑: 334↑: 335↑: 336↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 1123 1133 1213 1223 1313 1333 2003 2023 2113 2123 2203 2213 2223 2233 2323 2333 337↑: 338↑: 339↑: 340: 341: 342↑: 343↑: 344↑: 345↑: 346: 347↑: 348↑: 349: 350: 351↑: 352: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2223 2322 2322 2322 2322 2322 3003 3033 3113 3123 3133 3223 3233 3303 3313 3323 3323 2333 3003 3033 3233 3303 Symmetry 2021, 13, 1471 31 of 40

353↑: 354↑: 355: 356: 357: 358: 359: 360: 361↑: 362↑: 363: 364: 365↑: 366↑: 367: 368↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2322 2323 3020 3022 3023 3023 3121 3123 3123 3123 3222 3222 3222 3222 3222 3222 3333 3333 3333 3033 3033 3333 3333 1133 3133 3333 1313 1333 2333 3233 3313 3333 369↑: 370: 371: 372↑: 373↑: 374: 375: 376↑: 377↑: 378↑: 379: 380: 381: 382↑: 383: 384↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1112 1112 1112 1112 1112 1112 1112 1112 3223 3320 3321 3322 3322 3323 3323 3323 1222 1222 2020 2022 2220 2221 2222 2222 3333 3333 3333 2233 3333 0033 1133 3333 1333 3333 3333 3333 3333 3333 1333 2333 385↑: 386: 387↑: 388: 389↑: 390: 391: 392: 393↑: 394: 395↑: 396↑: 397↑: 398: 399↑: 400↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1112 1112 1112 1112 1113 1113 1113 1113 1113 1113 1113 1113 1113 1121 1121 1121 2222 2222 2222 2322 0020 0123 0222 1123 1123 2123 2222 2223 3123 2121 2122 2123 3033 3233 3333 3313 3333 3333 3333 3113 3333 3333 3133 3123 3333 1123 3333 3323 401↑: 402: 403: 404↑: 405↑: 406: 407: 408↑: 409: 410↑: 411↑: 412↑: 413↑: 414: 415↑: 416↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1121 1121 1121 1121 1121 1122 1122 1122 1122 1122 1123 1123 1131 1131 1131 1131 2222 2222 2222 2222 2222 1222 2122 2222 2222 2222 2122 2123 1222 1222 2022 2222 0003 0123 0333 1123 2123 1223 2123 2223 3223 3333 3133 3123 1333 3333 3333 1333 417: 418: 419: 420↑: 421↑: 422: 423↑: 424: 425: 426↑: 427: 428: 429↑: 430: 431: 432: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1131 1131 1131 1131 1131 1131 1131 1132 1133 1133 1133 1133 1133 2100 2100 2102 2222 2222 2222 2222 2222 2322 3222 2321 1323 2222 2323 3123 3323 2222 2222 2222 3003 3033 3303 3313 3333 3333 3333 3213 1333 3333 3333 3133 3333 2223 3333 0003 433: 434: 435: 436: 437: 438: 439↑: 440: 441: 442↑: 443↑: 444: 445: 446↑: 447: 448↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 2102 2110 2111 2111 2111 2111 2111 2111 2111 2111 2111 2111 2111 2111 2111 2111 2222 2220 1222 1222 1222 1222 1222 1322 2121 2221 2221 2222 2222 2222 2222 2222 3333 3333 1223 1333 2113 2333 3333 1233 2113 1333 3333 1333 2113 2233 2313 2333 449↑: 450: 451↑: 452↑: 453: 454↑: 455: 456↑: 457: 458: 459↑: 460: 461↑: 462↑: 463↑: 464: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 2111 2111 2111 2111 2111 2111 2111 2111 2111 2111 2112 2112 2112 2112 2112 2120 2222 2222 2222 2322 2322 3222 3222 3222 3222 3322 1221 2222 2222 2222 2222 2220 3233 3303 3333 2333 3333 1333 2333 3233 3333 3333 3333 1333 2333 3003 3333 3333 465: 466↑: 467: 468: 469: 470↑: 471↑: 472↑: 473: 474: 475↑: 476↑: 477↑: 478: 479↑: 480↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 2120 2121 2121 2121 2121 2121 2121 2121 2122 2122 2122 2122 2122 2122 2122 2122 2222 2222 2222 2222 2222 2222 2222 2222 0220 0220 2222 2222 2222 2222 2222 2222 3333 0123 0303 1113 1123 2123 2323 3333 0003 3333 0003 0223 0333 2003 2223 3223 481↑: 482: 483↑: 484↑: 485↑: 486: 487↑: 488: 489: 490: 491: 492: 493: 494: 495: 496: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 2122 2122 2123 2123 2123 2123 2123 2123 2131 2131 2132 2133 2133 2133 2133 3100 2222 3223 2222 2222 2222 2223 2223 3223 1222 2222 2222 2222 2222 2323 3222 2222 3333 3333 2323 3323 3333 3223 3333 3333 2313 2333 3333 2333 3333 2333 2333 3333 497: 498: 499: 500: 501: 502↑: 503: 504: 505: 506↑: 507: 508: 509: 510↑: 511↑: 512↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 3100 3101 3103 3103 3111 3111 3111 3111 3111 3111 3111 3111 3111 3111 3111 3111 3323 2222 2222 2222 1222 1222 1222 1323 2220 2222 2222 2222 2222 2222 2222 2322 3333 3303 3303 3333 1333 3313 3333 1333 3333 1333 2233 2333 3233 3313 3333 3333 513: 514: 515: 516: 517: 518: 519↑: 520↑: 521: 522: 523↑: 524: 525: 526: 527↑: 528↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 3111 3111 3111 3111 3111 3111 3111 3111 3112 3112 3113 3113 3113 3113 3113 3113 3121 3121 3121 3221 3222 3222 3222 3322 3221 3222 0123 0220 1121 1123 2222 3123 1333 3113 3333 3333 1333 2333 3333 3333 1333 3333 3333 3333 3333 3333 3333 3333 Symmetry 2021, 13, 1471 32 of 40

529↑: 530: 531: 532: 533: 534: 535↑: 536↑: 537↑: 538: 539↑: 540: 541: 542: 543↑: 544↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 3113 3122 3122 3122 3122 3123 3123 3123 3123 3123 3123 3130 3130 3131 3131 3131 3223 2222 3222 3222 3222 2222 2222 2222 2223 2323 3223 0020 2222 1222 2020 2222 3333 3333 2333 3223 3333 2323 3323 3333 3333 3323 3333 3333 3333 3333 3333 1313 545↑: 546↑: 547: 548: 549↑: 550↑: 551↑: 552: 553: 554↑: 555↑: 556: 557↑: 558: 559: 560↑: 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0001 0001 3131 3131 3132 3133 3133 3133 3133 3133 3133 3133 3133 3133 3133 0101 0103 1101 2222 3222 2222 0020 0020 0222 0323 2222 2222 2222 2323 3020 3323 0021 0021 2220 3333 3333 3333 0303 3333 3333 3333 0303 2323 3333 3333 3333 3333 3333 1313 3333 561↑: 562↑: 563↑: 564: 565: 566: 567: 568: 569↑: 570: 571↑: 572↑: 573: 574: 575↑: 576: 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 1101 1101 1101 1101 1110 1110 1110 1110 1110 1111 1111 1111 1111 1111 1111 1111 2222 2222 2322 3222 0021 1120 2220 2221 2222 0021 1221 1222 2021 2220 2221 2222 3303 3333 2333 3233 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 1333 577↑: 578: 579↑: 580↑: 581↑: 582↑: 583: 584: 585: 586↑: 587: 588: 589: 590: 591: 592: 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0001 0002 0002 0002 1111 1111 1111 1111 1112 1112 1112 1112 2110 2111 2111 2112 3111 0102 0102 0102 2222 2222 2222 2222 1222 2220 2221 2222 1220 2221 2222 2222 2222 0022 0023 2220 2333 3303 3313 3333 3333 3333 3333 3333 3333 3333 3333 3333 3033 3333 2233 3333 593: 594: 595↑: 596: 597: 598: 599: 600↑: 601↑: 602↑: 603: 604↑: 605: 606: 607↑: 608↑: 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0002 0102 1102 1110 1110 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 2222 2222 2022 2220 2222 1221 1222 2022 2022 2220 2221 2222 2222 2222 2222 2222 3333 3333 3333 3333 3333 3333 3333 3033 3333 3333 3333 1333 2333 3033 3233 3333 609: 610: 611: 612↑: 613↑: 614↑: 615↑: 616: 617: 618↑: 619↑: 620↑: 621: 622: 623↑: 624: 0002 0002 0002 0002 0002 0002 0002 0002 0002 0003 0003 0003 0003 0003 0003 0003 1111 1112 1112 1112 1131 2111 2112 2120 3111 0101 0103 0103 0103 0103 0103 0103 3222 2220 2221 2222 2022 2222 2222 2220 2022 0023 0022 0023 0323 2222 2222 2223 3303 3333 3333 3333 1333 3333 3333 3333 3313 3133 3323 3333 3333 0033 3333 0023 625↑: 626↑: 627↑: 628: 629↑: 630: 631: 632: 633: 634↑: 635↑: 636↑: 637: 638: 639↑: 640↑: 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0103 0111 0111 0111 0113 0113 0113 0113 0121 0122 0123 0123 0123 0123 0133 0133 2223 0121 0123 0222 0123 0223 2222 2223 2223 0222 0222 0223 2222 2223 0323 2222 3333 3113 3133 3123 3333 3333 3333 3333 3133 3223 3323 3333 3333 3333 3333 3333 641: 642↑: 643: 644: 645↑: 646↑: 647↑: 648↑: 649: 650↑: 651: 652: 653: 654: 655↑: 656↑: 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 0003 1111 1111 1111 1111 1111 1111 1111 1111 1113 1113 1113 1113 1113 1113 1113 1113 0023 0023 0123 0223 0323 1121 2222 2223 0023 0023 0122 0123 0223 1123 1123 2222 0303 3333 3333 3333 3333 3333 0333 0323 0103 3333 3323 3333 3333 0113 3333 0133 657↑: 658↑: 659↑: 660↑: 661: 662↑: 663: 664: 665: 666: 667: 668: 669↑: 670↑: 671↑: 672↑: 0003 0003 0003 0003 0003 0003 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 1113 1121 1123 2122 2123 2123 0110 0120 1101 1101 1101 1101 1101 1110 1110 1110 2223 2122 2123 2222 2223 2223 2222 1221 2222 2222 2222 2222 2222 2220 2222 2222 0123 0333 0123 3333 0223 3333 0013 0013 0013 1103 3303 3313 3333 3333 3303 3333 673↑: 674: 675: 676↑: 677: 678↑: 679↑: 680: 681: 682↑: 683↑: 684↑: 685↑: 686↑: 687: 688: 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 0010 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1131 1131 1131 1131 3222 1222 2220 2221 2222 2222 2222 2222 2222 2222 2222 3222 2222 2222 2222 2222 3233 3333 3333 3333 0013 1313 1333 3013 3303 3313 3333 3333 1333 3303 3313 3333 689: 690: 691↑: 692: 693: 694: 695: 696↑: 697↑: 698: 699↑: 700↑: 701: 702↑: 703↑: 704↑: 0010 0010 0010 0010 0010 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 2111 3101 3111 3111 3131 0122 0133 1100 1100 1111 1111 1111 1111 1111 1111 1111 2022 2222 2222 2222 2222 1222 1323 2222 3322 2220 2221 2222 2222 2222 2222 3322 3333 1303 3313 3333 3333 1223 1333 3333 2233 3333 3333 2233 3303 3313 3333 3333 Symmetry 2021, 13, 1471 33 of 40

705: 706: 707↑: 708↑: 709↑: 710: 711: 712↑: 713: 714↑: 715↑: 716↑: 717: 718↑: 719: 720: 0012 0012 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 1102 1112 0110 0111 0111 0111 0113 0120 0120 0120 0120 0120 0120 0121 0121 0121 2220 2222 2122 2121 2122 2123 2122 2222 2222 2222 2222 2223 3323 2222 2222 2222 3333 3333 0023 0113 0123 0333 3323 0023 0223 3323 3333 0033 0023 0123 0323 3323 721: 722↑: 723↑: 724↑: 725: 726: 727↑: 728↑: 729: 730↑: 731↑: 732↑: 733↑: 734↑: 735↑: 736: 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0020 0121 0122 0122 0123 0123 0123 0123 0133 1111 1111 1111 1111 1111 1111 1113 1121 2222 2222 2222 2222 2222 2222 2223 2323 0220 0222 0223 2222 2222 2222 0222 0120 3333 0223 3333 0323 3323 3333 0333 0333 0023 3333 3323 0023 0223 1113 3133 0023 737: 738↑: 739↑: 740↑: 741↑: 742↑: 743: 744↑: 745↑: 746: 747: 748↑: 749: 750: 751↑: 752↑: 0020 0020 0020 0020 0020 0020 0020 0020 0020 0022 0022 0022 0022 0022 0022 0022 1121 1121 1121 1121 1121 1123 3123 3123 3133 0100 0100 0111 0120 0120 0122 0122 0121 0122 0123 2222 2222 0123 0323 2222 2222 2022 2222 2122 2222 2222 2222 2222 1123 3333 3323 0023 1123 3123 3323 3323 3333 2023 3333 2123 2023 3333 2223 3333 753↑: 754↑: 755: 756: 757↑: 758: 759↑: 760: 761↑: 762↑: 763: 764↑: 765↑: 766↑: 767: 768↑: 0022 0022 0022 0022 0022 0022 0022 0022 0022 0022 0023 0023 0023 0023 0023 0023 0123 0123 1111 1111 1111 1111 1111 1122 1122 1133 0111 0113 0121 0123 0123 0123 2222 2222 0222 2022 2222 2222 2222 2222 2222 0022 2122 2123 2222 2222 2222 2223 2323 3333 0223 2023 2223 2323 3333 2223 3333 1133 3133 3333 3123 3323 3333 3333 769: 770: 771↑: 772: 773: 774↑: 775↑: 776↑: 777↑: 778↑: 779↑: 780↑: 781: 782↑: 783: 784: 0023 0023 0023 0023 0023 0023 0023 0023 0023 0023 0023 0023 0030 0030 0030 0030 0123 0123 0123 0123 0123 0123 1111 1111 1113 1121 1123 1123 0130 1101 1101 1101 3023 3023 3223 3320 3323 3323 0222 0223 0223 0123 0122 0123 3322 2222 2222 2222 3023 3323 3323 0023 0023 3323 0333 0323 0123 0323 0133 0123 0023 3033 3303 3333 785↑: 786↑: 787: 788: 789: 790↑: 791↑: 792: 793↑: 794↑: 795: 796↑: 797: 798: 799: 800↑: 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 0030 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1112 1130 1131 1131 1131 1222 2022 2022 2222 2222 2222 2222 2222 2222 2322 3222 1222 2222 2222 2222 2222 3333 3033 3333 1313 1333 3033 3303 3313 3333 3333 3333 3033 3333 3303 3313 3333 801: 802: 803↑: 804↑: 805: 806: 807: 808: 809: 810: 811↑: 812↑: 813↑: 814: 815↑: 816: 0030 0030 0030 0030 0031 0031 0032 0032 0033 0033 0033 0033 0033 0033 0033 0033 2111 3103 3111 3131 1130 1131 0132 1111 0100 0103 0111 0123 0133 1111 1111 1111 2221 2222 2222 2222 2222 2222 3320 3220 3023 3023 3123 3223 3323 0323 2222 3023 3033 3303 3333 3333 3303 3333 2203 2303 3033 3333 3133 3333 3333 0333 3333 3033 817: 818↑: 819↑: 820↑: 821: 822↑: 823: 824↑: 825↑: 826↑: 827↑: 828↑: 829↑: 830↑: 831↑: 832↑: 0033 0033 0033 0033 0033 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 0100 1111 1111 1122 1133 1133 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 3223 3323 1122 2222 3323 0120 0120 0120 0120 0121 0121 2122 2122 2222 2222 2222 3333 3333 0033 3333 3333 0103 0113 3133 3333 0113 3333 0103 2123 0103 0113 2223 833↑: 834↑: 835↑: 836↑: 837↑: 838↑: 839: 840: 841: 842: 843: 844: 845: 846↑: 847↑: 848↑: 0100 0100 0100 0100 0100 0100 0100 0100 0101 0101 0101 0101 0101 0101 0101 0101 1111 1111 1113 1121 1122 1133 2122 3133 1101 1101 1111 1111 1111 1111 1111 1111 3123 3323 0120 0220 0222 0323 0120 0120 0020 2222 0020 0120 0120 0121 0121 0123 3133 3333 0303 0103 0223 0333 0103 0103 1103 1103 3333 1103 3333 1113 3333 1133 849↑: 850: 851↑: 852↑: 853: 854↑: 855↑: 856: 857: 858↑: 859: 860: 861↑: 862: 863: 864: 0101 0101 0101 0101 0101 0101 0101 0103 0103 0103 0103 0103 0103 0110 0110 0110 1111 1111 1111 1111 1111 1111 1121 3103 3103 3113 3130 3133 3133 1110 1110 1111 0123 2121 2121 2222 2222 2222 0222 0123 0123 0123 0123 0123 0123 1120 1120 1120 3333 1113 3333 1113 1133 3333 1123 3103 3133 3133 0103 0103 3133 0003 3333 0103 865↑: 866: 867↑: 868↑: 869: 870: 871: 872↑: 873↑: 874↑: 875↑: 876↑: 877: 878: 879↑: 880: 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0111 0111 0111 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1113 1111 1111 1111 1111 1121 1121 1121 1122 1122 2222 2222 2222 2222 2222 2222 1123 1120 1120 1121 1121 0113 3113 3333 0123 3333 0003 0103 0113 0123 3113 3333 0333 1103 1113 1113 2113 Symmetry 2021, 13, 1471 34 of 40

881↑: 882↑: 883: 884: 885: 886↑: 887↑: 888↑: 889↑: 890↑: 891↑: 892: 893↑: 894↑: 895↑: 896↑: 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1121 1122 1122 1122 1122 1122 1123 1123 2222 2222 2222 2222 2222 2222 2222 2223 3333 1123 1133 2123 2223 3333 1133 3333 1113 1123 1133 2213 2223 3323 3333 3333 897: 898: 899: 900: 901↑: 902: 903: 904: 905: 906: 907: 908: 909: 910: 911: 912: 0111 0111 0111 0111 0111 0111 0111 0111 0111 0112 0112 0112 0112 0112 0112 0113 1111 1111 1111 1111 1111 1122 1122 1133 1133 1111 1111 1112 1112 1112 1112 1110 3121 3123 3321 3323 3323 1222 2222 1323 2222 1121 2222 1122 1123 2221 2222 1120 1113 1133 3333 1133 3333 1223 3333 1333 3333 2113 3333 3333 2233 3333 3333 3003 913↑: 914↑: 915↑: 916↑: 917↑: 918↑: 919: 920: 921: 922↑: 923↑: 924: 925↑: 926: 927↑: 928: 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 0113 1111 1111 1111 1111 1111 1111 1111 1111 1111 1113 1113 1113 1113 1113 1113 1113 1121 1121 1123 2222 2222 2222 3121 3123 3123 1122 1123 2222 2222 2223 2223 3123 3113 3333 3133 3113 3133 3333 3113 3133 3333 3323 3333 1133 3333 1123 3333 3333 929: 930: 931: 932: 933↑: 934↑: 935: 936: 937: 938: 939↑: 940: 941↑: 942: 943↑: 944: 0113 0113 0113 0113 0113 0113 0113 0120 0120 0120 0120 0120 0120 0121 0121 0121 1133 1133 3111 3111 3113 3123 3133 2102 2120 2120 2121 2122 2122 1101 1111 1111 1323 2222 3121 3121 3123 3123 1321 0120 2120 2122 2122 0120 2122 2020 2121 2121 3333 3333 1333 3113 3113 0113 3113 0123 0123 0123 0123 0123 0123 1103 1113 2113 945↑: 946↑: 947: 948↑: 949↑: 950: 951↑: 952↑: 953: 954↑: 955↑: 956↑: 957: 958: 959: 960: 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 0121 1111 1111 1111 1111 1111 1111 1111 1121 1121 1121 1121 1121 1121 1122 1122 2111 2121 2122 2122 2122 2222 2222 2222 2222 2222 2222 2222 2223 3323 2222 2222 1222 3333 1123 2123 3333 1113 2123 3333 1123 2123 3323 3333 1133 1123 1223 3333 2113 961: 962↑: 963: 964↑: 965: 966: 967↑: 968↑: 969↑: 970↑: 971↑: 972↑: 973↑: 974↑: 975↑: 976↑: 0121 0121 0121 0121 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 2111 2121 2122 2123 1100 1111 1111 1111 1111 1111 1111 1113 1121 1121 1122 1122 2121 2121 2121 0121 2022 2121 2122 2122 2122 2222 2222 2122 2222 2222 2222 2222 2113 2123 1213 2123 2023 3333 2123 3133 3333 2223 3333 2323 2123 3333 2223 3333 977: 978: 979↑: 980: 981↑: 982↑: 983↑: 984↑: 985: 986↑: 987: 988: 989: 990↑: 991↑: 992↑: 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 0122 0123 0123 0123 2100 2111 2121 2121 2122 2122 2123 2123 2123 2133 3122 3123 3133 1111 1111 1111 0122 1222 2122 2122 0123 2122 0123 2122 2122 0123 0123 0123 2122 2121 2121 2222 0123 1223 2123 3333 0123 2123 0123 2123 3133 0123 0123 0123 2123 3113 3333 3113 993↑: 994↑: 995↑: 996: 997↑: 998↑: 999↑: 1000↑: 1001: 1002↑: 1003↑: 1004: 1005: 1006↑: 1007↑: 1008: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 1111 1111 1111 1121 1121 1122 1122 1123 1123 1123 1123 1123 1123 1123 1133 2100 2222 2323 3123 3121 3123 2222 2222 2323 3123 3123 3123 3321 3323 3323 2323 0123 3223 3333 3123 3123 3123 3223 3333 3323 1123 3123 3323 1123 1123 3323 3333 0123 1009: 1010: 1011↑: 1012: 1013↑: 1014: 1015↑: 1016: 1017: 1018↑: 1019: 1020↑: 1021↑: 1022↑: 1023↑: 1024: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 2102 2102 2103 2103 2103 2113 2120 2120 2120 2121 2122 2122 2122 2122 2122 2122 0123 3120 0123 0123 0123 2123 0123 0123 0123 2123 0120 0123 0123 2122 2122 3123 0123 0123 0123 1013 2103 2113 0103 0123 2123 2123 0123 0123 2123 0123 3123 0123 1025: 1026↑: 1027: 1028↑: 1029↑: 1030↑: 1031: 1032: 1033↑: 1034↑: 1035: 1036: 1037↑: 1038↑: 1039: 1040: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 2122 2123 2123 2123 2123 2123 2123 2123 2123 2123 2123 2123 2123 2123 2123 2123 3123 0123 0123 0123 0123 0123 0123 0123 0123 0123 1123 1123 2122 2123 2123 2123 3123 0003 0023 0103 0123 0223 1113 2023 2123 2223 1123 2123 3123 0123 1113 1123 1041: 1042↑: 1043↑: 1044↑: 1045↑: 1046↑: 1047: 1048: 1049↑: 1050: 1051: 1052: 1053: 1054: 1055: 1056↑: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 2123 2123 2123 2123 2123 2123 2123 2123 2130 2132 2132 2133 2133 2133 3100 3103 2123 2123 2123 2123 2123 2123 3123 3123 0123 0123 3123 0123 2123 2123 0123 0120 1213 2123 2323 3123 3133 3333 2123 3123 0123 0123 3123 0123 2133 3133 0123 0123 Symmetry 2021, 13, 1471 35 of 40

1057: 1058: 1059↑: 1060↑: 1061: 1062↑: 1063: 1064: 1065: 1066: 1067↑: 1068: 1069↑: 1070↑: 1071↑: 1072↑: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 3103 3103 3113 3120 3120 3120 3121 3122 3122 3122 3123 3123 3123 3123 3123 3123 0123 3123 3123 0123 1021 3120 3121 0123 2122 3122 0020 0023 0120 0123 0123 0323 0123 0123 3123 0123 0123 0123 3123 0123 3123 3123 0123 0123 0123 0123 3123 0123 1073: 1074: 1075: 1076: 1077: 1078↑: 1079: 1080↑: 1081↑: 1082↑: 1083: 1084↑: 1085: 1086: 1087↑: 1088↑: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 3123 1121 1121 1123 1123 1321 2122 2123 2123 2123 2222 3023 3123 3123 3123 3123 3223 0123 3123 1123 3123 3123 3123 2123 3123 3133 3123 0123 0123 1123 2123 3123 3123 1089↑: 1090: 1091: 1092: 1093: 1094: 1095↑: 1096: 1097↑: 1098: 1099↑: 1100↑: 1101: 1102: 1103: 1104: 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0123 0131 0131 0132 0132 3123 3130 3130 3132 3132 3133 3133 3133 3133 3133 3133 3133 1111 1131 1132 2100 3323 0123 0123 0123 2123 0123 0123 0123 0123 2123 2123 3123 3121 3322 3321 0123 0123 0123 2103 0123 2123 0103 0123 2123 3133 2123 3133 0123 1113 1123 2213 0123 1105: 1106↑: 1107↑: 1108↑: 1109↑: 1110↑: 1111↑: 1112: 1113: 1114: 1115: 1116: 1117: 1118↑: 1119↑: 1120: 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 0133 1100 1111 1111 1113 1121 1133 2122 2122 2123 2133 3100 3111 3113 3113 3123 3123 3023 3123 3323 3123 3223 3323 0123 3123 0123 0123 0123 1323 2222 3123 0123 2122 3033 3133 3333 3333 3133 3333 0123 3133 0123 0123 0133 1333 3133 3133 0123 3133 1121↑: 1122↑: 1123↑: 1124: 1125: 1126↑: 1127↑: 1128: 1129: 1130↑: 1131↑: 1132: 1133: 1134: 1135: 1136: 0133 0133 0133 0200 0200 0200 0200 0202 0202 0202 0202 0210 0210 0211 0212 0213 3123 3133 3133 2111 2112 3111 3111 2111 2133 3131 3131 2101 2102 2100 2101 2132 3123 0123 3123 2222 0120 2220 2222 0121 0323 2020 2222 1022 1021 1022 1020 1321 3133 0123 3133 3333 0203 3313 3333 2113 2333 1313 3333 3333 0213 1023 2103 3213 1137↑: 1138: 1139: 1140: 1141↑: 1142: 1143↑: 1144: 1145↑: 1146↑: 1147↑: 1148: 1149: 1150: 1151: 1152: 0213 0220 0220 0220 0220 0220 0220 0220 0220 0220 0220 0221 0221 0221 0221 0221 3120 2120 2120 2120 2121 2121 2122 2122 2122 2123 3113 2121 2121 2121 2122 2122 3120 2220 2220 2222 2222 2222 2222 2222 2222 2223 0220 2221 2221 2222 2222 2222 0213 0003 3333 0023 0123 3333 0223 3223 3333 0333 3113 1113 3333 1123 1223 2223 1153: 1154: 1155: 1156↑: 1157: 1158↑: 1159↑: 1160↑: 1161↑: 1162↑: 1163: 1164: 1165: 1166: 1167: 1168↑: 0221 0222 0222 0222 0222 0222 0222 0222 0222 0222 0222 0222 0222 0222 0222 0222 2123 2120 2120 2121 2121 2121 2122 2122 2123 2123 2133 3122 3123 3132 3133 3133 2221 2222 2222 2222 2222 2222 2222 2222 2222 2222 2323 2222 2222 2222 2222 2222 1313 2023 2223 2123 2323 3333 2223 3333 2323 3333 2333 2223 2323 3333 2323 3333 1169: 1170↑: 1171↑: 1172↑: 1173↑: 1174↑: 1175: 1176: 1177: 1178: 1179: 1180↑: 1181↑: 1182: 1183: 1184: 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 0223 2120 2121 2122 2122 2123 2123 2132 2133 3122 3122 3122 3123 3123 3123 3123 3123 2220 2223 2222 2222 2222 2223 3323 2323 2222 3222 3222 0120 0123 2222 2222 2223 3003 3133 3223 3333 3323 3333 3223 3333 3223 2333 3223 0113 0123 3323 3333 3333 1185↑: 1186↑: 1187↑: 1188: 1189: 1190: 1191: 1192: 1193: 1194: 1195: 1196: 1197: 1198: 1199: 1200: 0223 0223 0231 0232 0232 0233 0233 0233 0233 0233 0300 0300 0301 0301 0303 0303 3123 3123 3102 2103 2123 2112 2122 2123 2133 3133 3111 3131 3110 3130 3103 3103 3123 3223 1320 3020 3323 3123 3223 3223 3323 3323 2222 0320 2222 0321 0020 0023 0223 3223 2013 2303 2323 3233 3233 3333 3333 3333 3333 0103 1033 1013 3303 3333 1201: 1202↑: 1203: 1204↑: 1205↑: 1206↑: 1207: 1208: 1209: 1210: 1211: 1212: 1213: 1214: 1215: 1216: 0303 0303 0303 0303 0303 0303 0303 0311 0313 0313 0313 0313 0313 0313 0321 0322 3103 3113 3113 3123 3133 3133 3133 3100 3113 3113 3113 3123 3133 3133 3132 3122 2222 0123 2222 0222 0323 2222 2323 1023 1121 1123 2222 1221 1323 3323 2321 2222 3303 3333 3333 3323 3333 3333 3333 1033 3313 3333 3313 3313 3333 3333 1213 2223 1217: 1218: 1219↑: 1220↑: 1221↑: 1222↑: 1223↑: 1224↑: 1225↑: 1226↑: 1227: 1228: 1229: 1230: 1231: 1232: 0322 0323 0323 0323 0323 0323 0323 0323 0323 0323 0330 0330 0331 0332 0333 0333 3131 3103 3113 3123 3123 3123 3123 3123 3123 3133 3111 3122 3110 3120 3103 3103 2322 2020 2122 2222 2223 2323 3123 3223 3323 2323 3121 3222 3120 3220 3023 3323 2123 3303 3323 3323 3333 3323 3323 3323 3323 3333 0113 0223 1003 2003 3333 3333 Symmetry 2021, 13, 1471 36 of 40

1233↑: 1234: 1235↑: 1236↑: 0333 0333 0333 0333 3113 3113 3123 3133 3123 3223 3223 3323 3333 3333 3333 3333

Appendix C ∗(2) s Witnesses gi ∈ C4 for maximal centralising monoids, given as conjugates gj (with s ∈ Sym(4)) of the representatives from AppendixA (for the indices i, j see AppendixB). (01) (210) (0321) g1 : g879 = g1 , g1159 = g1 , g1236 = g1 (23) (123) (01) (01)(23) (0123) g4 : g15 = g4 , g182 = g4 , g881 = g4 , g889 = g4 , g240 = g4 , (210) (023) (03) g1160 = g4 , g479 = g4 , g557 = g4 (23) (12) (123) (321) (13) (01) g6 : g11 = g6 , g26 = g6 , g42 = g6 , g51 = g6 , g105 = g6 , g882 = g6 , (01)(23) (012) (0123) (0132) (013) g887 = g6 , g865 = g6 , g913 = g6 , g846 = g6 , g943 = g6 , (210) (0231) (02) (023) (02)(13) g1156 = g6 , g1161 = g6 , g1143 = g6 , g1171 = g6 , g751 = g6 , (0213) (0321) (310) (320) (03) g975 = g6 , g1233 = g6 , g1235 = g6 , g1205 = g6 , g1226 = g6 , (0312) (03)(12) g813 = g6 , g1110 = g6 (23) (12) (123) (321) (13) (01) g10 : g17 = g10 , g30 = g10 , g184 = g10 , g61 = g10 , g214 = g10 , g886 = g10 , (01)(23) (012) (0123) (0132) (013) g891 = g10 , g867 = g10 , g241 = g10 , g852 = g10 , g281 = g10 , (210) (02) (023) (02)(13) (0213) g1158 = g10 , g1145 = g10 , g480 = g10 , g757 = g10 , g408 = g10 , (320) (03) (0312) (03)(12) g1206 = g10 , g555 = g10 , g818 = g10 , g429 = g10 (23) (12) (123) (321) (13) (01) g12 : g16 = g12 , g43 = g12 , g183 = g12 , g111 = g12 , g200 = g12 , g888 = g12 , (01)(23) (012) (0123) (0132) (013) g890 = g12 , g914 = g12 , g237 = g12 , g949 = g12 , g203 = g12 , (210) (02) (023) (02)(13) (0213) g1162 = g12 , g1172 = g12 , g476 = g12 , g970 = g12 , g126 = g12 , (03) (0312) (03)(12) g551 = g12 , g1107 = g12 , g150 = g12 (23) (12) (123) (321) (13) (01) g19 : g25 = g19 , g186 = g19 , g549 = g19 , g236 = g19 , g475 = g19 , g893 = g19 , (01)(23) (0231) g901 = g19 , g1168 = g19 (23) (12) (123) (321) (13) (01) g20 : g22 = g20 , g187 = g20 , g389 = g20 , g284 = g20 , g401 = g20 , g894 = g20 , (01)(23) (012) (0123) (0132) (013) g896 = g20 , g238 = g20 , g646 = g20 , g222 = g20 , g734 = g20 , (02) (023) (02)(13) (0213) (320) g477 = g20 , g660 = g20 , g92 = g20 , g832 = g20 , g550 = g20 , (03) (0312) (03)(12) g745 = g20 , g96 = g20 , g834 = g20 (12) (321) (01) (012) (0132) (02) g21 : g188 = g21 , g304 = g21 , g895 = g21 , g242 = g21 , g320 = g21 , g481 = g21 , (02)(13) (320) (0312) g333 = g21 , g554 = g21 , g376 = g21 (23) (123) (01) (01)(23) (023) (0123) g31 : g55 = g31 , g128 = g31 , g868 = g31 , g848 = g31 , g998 = g31 , g990 = g31 , (210) (0231) (0321) (310) (03) g1141 = g31 , g753 = g31 , g1202 = g31 , g812 = g31 , g1007 = g31 (23) (12) (123) (321) (13) (01) g33 : g53 = g33 , g190 = g33 , g107 = g33 , g197 = g33 , g48 = g33 , g872 = g33 , (01)(23) (012) (0123) (0132) (013) g847 = g33 , g245 = g33 , g945 = g33 , g250 = g33 , g916 = g33 , (0231) (02) (023) (02)(13) (320) g752 = g33 , g470 = g33 , g976 = g33 , g483 = g33 , g528 = g33 , (0312) g539 = g33 (23) (12) (123) (321) (13) (01) g35 : g56 = g35 , g192 = g35 , g129 = g35 , g208 = g35 , g136 = g35 , g873 = g35 , (01)(23) (012) (0123) (0132) (013) g849 = g35 , g244 = g35 , g991 = g35 , g211 = g35 , g992 = g35 , (0231) (02) (023) (02)(13) (0213) g754 = g35 , g466 = g35 , g999 = g35 , g139 = g35 , g993 = g35 , (320) (0312) (03)(12) g523 = g35 , g143 = g35 , g994 = g35 (23) (12) (01) (01)(23) (012) g36 : g58 = g36 , g191 = g36 , g874 = g36 , g851 = g36 , g246 = g36 , (0231) (02) (310) (320) g761 = g36 , g471 = g36 , g820 = g36 , g529 = g36 Symmetry 2021, 13, 1471 37 of 40

(23) (12) (123) (321) (13) (01) g39 : g63 = g39 , g196 = g39 , g216 = g39 , g307 = g39 , g314 = g39 , g875 = g39 , (01)(23) (012) (0123) (0132) (013) g854 = g39 , g249 = g39 , g283 = g39 , g322 = g39 , g339 = g39 , (0231) (02) (023) (02)(13) (0213) g759 = g39 , g472 = g39 , g410 = g39 , g335 = g39 , g342 = g39 , (310) (320) (03) (0312) (03)(12) g815 = g39 , g527 = g39 , g426 = g39 , g369 = g39 , g354 = g39 (23) (123) (01) (01)(23) (03) (0123) g44 : g108 = g44 , g65 = g44 , g915 = g44 , g946 = g44 , g639 = g44 , g826 = g44 , (210) (0231) (023) (0321) (310) g1173 = g44 , g973 = g44 , g722 = g44 , g1221 = g44 , g1108 = g44 (23) (12) (123) (321) (13) (01) g49 : g110 = g49 , g198 = g49 , g67 = g49 , g217 = g49 , g87 = g49 , g917 = g49 , (01)(23) (012) (0123) (0132) (013) g948 = g49 , g251 = g49 , g827 = g49 , g287 = g49 , g831 = g49 , (0231) (02) (023) (02)(13) (0213) g974 = g49 , g484 = g49 , g723 = g49 , g405 = g49 , g733 = g49 , (320) (03) (0312) (03)(12) g537 = g49 , g640 = g49 , g397 = g49 , g645 = g49 (23) (12) (123) (321) (13) (01) g50 : g113 = g50 , g199 = g50 , g202 = g50 , g306 = g50 , g308 = g50 , g918 = g50 , (01)(23) (012) (0123) (0132) (013) g951 = g50 , g252 = g50 , g204 = g50 , g321 = g50 , g309 = g50 , (0231) (02) (023) (02)(13) (0213) g971 = g50 , g485 = g50 , g127 = g50 , g330 = g50 , g313 = g50 , (320) (03) (0312) (03)(12) g536 = g50 , g149 = g50 , g362 = g50 , g230 = g50 (12) (321) (01) (012) (0132) (210) g68 : g123 = g68 , g146 = g68 , g822 = g68 , g967 = g68 , g1106 = g68 , g712 = g68 , (02) (02)(13) (0321) (320) (0312) g952 = g68 , g1220 = g68 , g620 = g68 , g923 = g68 , g1174 = g68 (23) (12) (123) (13) (01) (01)(23) g69 : g78 = g69 , g124 = g69 , g81 = g69 , g86 = g69 , g824 = g69 , g828 = g69 , (012) (0123) (013) (210) (0231) g968 = g69 , g829 = g69 , g833 = g69 , g714 = g69 , g740 = g69 , (02) (023) (0213) (0321) (310) g954 = g69 , g741 = g69 , g744 = g69 , g625 = g69 , g650 = g69 , (320) (03) (03)(12) g927 = g69 , g655 = g69 , g662 = g69 (23) (12) (123) (321) (13) (01) g70 : g88 = g70 , g125 = g70 , g219 = g70 , g148 = g70 , g229 = g70 , g825 = g70 , (01)(23) (012) (0123) (013) (210) g830 = g70 , g969 = g70 , g288 = g70 , g361 = g70 , g715 = g70 , (0231) (02) (023) (0213) (0321) g732 = g70 , g955 = g70 , g404 = g70 , g535 = g70 , g623 = g70 , (310) (320) (03) (03)(12) g642 = g70 , g925 = g70 , g393 = g70 , g487 = g70 (23) (12) (123) (321) (13) (210) g71 : g75 = g71 , g114 = g71 , g130 = g71 , g98 = g71 , g133 = g71 , g718 = g71 , (0231) (02)(13) (0321) (310) (0312) g724 = g71 , g766 = g71 , g629 = g71 , g636 = g71 , g768 = g71 (23) (210) (0321) g97 : g118 = g97 , g765 = g97 , g764 = g97 (01) (210) (0321) g135 : g995 = g135 , g1082 = g135 , g1046 = g135 (23) (12) (123) (321) (13) (01) g151 : g154 = g151 , g172 = g151 , g270 = g151 , g164 = g151 , g276 = g151 , g699 = g151 , (01)(23) (012) (0123) (0132) (013) g702 = g151 , g571 = g151 , g253 = g151 , g678 = g151 , g259 = g151 , (02) (023) (0213) (320) (03) g615 = g151 , g461 = g151 , g446 = g151 , g804 = g151 , g546 = g151 , (03)(12) g520 = g151 (23) (12) (123) (321) (13) (01) g153 : g157 = g153 , g269 = g153 , g543 = g153 , g255 = g153 , g462 = g153 , g700 = g153 , (01)(23) (0231) g704 = g153 , g1131 = g153 (12) (321) (01) (012) (0132) (02) g156 : g272 = g156 , g337 = g156 , g703 = g156 , g256 = g156 , g325 = g156 , g463 = g156 , (02)(13) (320) (0312) g334 = g156 , g545 = g156 , g373 = g156 (23) (12) (123) (321) (13) (01) g159 : g178 = g159 , g173 = g159 , g294 = g159 , g165 = g159 , g279 = g159 , g676 = g159 , (01)(23) (012) (0123) (0132) (013) g579 = g159 , g572 = g159 , g299 = g159 , g679 = g159 , g263 = g159 , (0231) (02) (023) (0213) (310) g607 = g159 , g614 = g159 , g384 = g159 , g449 = g159 , g794 = g159 , (320) (03) (03)(12) g803 = g159 , g423 = g159 , g512 = g159 (23) (12) (01) (01)(23) (012) g166 : g174 = g166 , g277 = g166 , g682 = g166 , g575 = g166 , g260 = g166 , (0231) (02) (310) (320) g612 = g166 , g448 = g166 , g800 = g166 , g519 = g166 Symmetry 2021, 13, 1471 38 of 40

(23) (12) (123) (321) (13) (01) g168 : g180 = g168 , g280 = g168 , g298 = g168 , g338 = g168 , g344 = g168 , g683 = g168 , (01)(23) (012) (0123) (0132) (013) g580 = g168 , g264 = g168 , g303 = g168 , g326 = g168 , g345 = g168 , (0231) (02) (023) (02)(13) (0213) g608 = g168 , g451 = g168 , g387 = g168 , g336 = g168 , g343 = g168 , (310) (320) (03) (0312) (03)(12) g793 = g168 , g511 = g168 , g421 = g168 , g368 = g168 , g353 = g168 (23) (12) (123) (321) (13) (01) g169 : g177 = g169 , g415 = g169 , g297 = g169 , g385 = g169 , g301 = g169 , g684 = g169 , (01)(23) (012) (0132) (210) g577 = g169 , g785 = g169 , g604 = g169 , g1127 = g169 (12) (321) (012) (0132) g347 : g395 = g347 , g399 = g347 , g647 = g347 , g730 = g347 (23) (12) (123) (321) (13) g348 : g366 = g348 , g416 = g348 , g510 = g348 , g378 = g348 , g443 = g348 , (012) (0132) (02)(13) (0213) g790 = g348 , g601 = g348 , g562 = g348 , g672 = g348 (12) (321) (01) (012) (0132) g351 : g420 = g351 , g382 = g351 , g365 = g351 , g791 = g351 , g602 = g351 , (210) (02)(13) (0321) (0312) g506 = g351 , g569 = g351 , g439 = g351 , g669 = g351 (12) (321) (02)(13) g372 : g544 = g372 , g459 = g372 , g696 = g372 (23) (123) (01) (01)(23) (210) g377 : g413 = g377 , g452 = g377 , g600 = g377 , g786 = g377 , g561 = g377 , (023) (0321) (03) g691 = g377 , g670 = g377 , g586 = g377 (23) (123) (01) (01)(23) (0123) g396 : g400 = g396 , g411 = g396 , g648 = g396 , g731 = g396 , g775 = g396 , (210) (0321) g656 = g396 , g738 = g396 (01) g412 : g776 = g412 (23) (123) (210) (0231) (023) g442 : g502 = g442 , g456 = g442 , g671 = g442 , g782 = g442 , g685 = g442 , (0321) (310) (03) g560 = g442 , g595 = g442 , g581 = g442 (210) (0321) g454 : g686 = g454 , g582 = g454 (23) (12) (123) (321) g563 : g673 = g563 , g613 = g563 , g1126 = g563 , g796 = g563 (23) (12) (123) (321) (13) (01) g618 : g707 = g618 , g619 = g618 , g836 = g618 , g716 = g618 , g835 = g618 , g876 = g618 , (01)(23) (012) (0123) (0132) (013) g855 = g618 , g922 = g618 , g708 = g618 , g956 = g618 , g626 = g618 , (210) (0231) (02) (023) (02)(13) g1146 = g618 , g748 = g618 , g1170 = g618 , g837 = g618 , g972 = g618 , (0213) (0321) (310) (320) (03) g634 = g618 , g1204 = g618 , g811 = g618 , g1219 = g618 , g838 = g618 , (0312) (03)(12) g1109 = g618 , g728 = g618 (23) (12) (321) g627 : g709 = g627 , g635 = g627 , g727 = g627 (23) g657 : g739 = g657 (23) g658 : g735 = g658 (23) (01) (01)(23) g659 : g742 = g659 , g777 = g659 , g778 = g659 (12) g697 : g1130 = g697 (23) (13) g762 : g819 = g762 , g1147 = g762 (12) (321) (01) g771 : g858 = g771 , g939 = g771 , g1000 = g771 (12) (321) (01) (210) (0321) g774 : g861 = g774 , g941 = g774 , g1006 = g774 , g1099 = g774 , g1023 = g774 g779 : no conjugates needed to represent all maximal centralising monoids g780 : no conjugates needed to represent all maximal centralising monoids (23) (12) g933 : g962 = g933 , g1186 = g933 (23) (12) g934 : g964 = g934 , g1185 = g934 (23) (123) (0123) g979 : g1118 = g979 , g1224 = g979 , g1222 = g979 (23) (0123) (013) (023) (03) g981 : g1122 = g981 , g1089 = g981 , g1034 = g981 , g1100 = g981 , g1021 = g981 (23) (123) g982 : g1123 = g982 , g1225 = g982 (23) (0123) (013) g983 : g1119 = g983 , g1072 = g983 , g1030 = g983 Symmetry 2021, 13, 1471 39 of 40

(23) (123) g984 : g1121 = g984 , g1223 = g984 (02) (320) g986 : g1056 = g986 , g1015 = g986 (12) (321) g997 : g1078 = g997 , g1045 = g997 (12) (321) g1002 : g1080 = g1002, g1044 = g1002 (12) (321) g1003 : g1081 = g1003, g1037 = g1003 (23) g1011 : g1060 = g1011 (23) g1013 : g1062 = g1013 (23) (123) (13) g1018 : g1059 = g1018, g1088 = g1018 , g1043 = g1018 (23) g1020 : g1095 = g1020 (23) g1022 : g1097 = g1022 (23) (0231) g1026 : g1067 = g1026, g1111 = g1026 (23) g1028 : g1069 = g1028 (23) g1029 : g1070 = g1029 (23) g1033 : g1084 = g1033 (23) g1038 : g1071 = g1038 (23) g1042 : g1087 = g1042 g1049 : no conjugates needed to represent all maximal centralising monoids g1137 : no conjugates needed to represent all maximal centralising monoids g1180 : no conjugates needed to represent all maximal centralising monoids g1181 : no conjugates needed to represent all maximal centralising monoids g1187 : no conjugates needed to represent all maximal centralising monoids

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