The Endotopism Semigroups of an Equivalence Relation

1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References

The endotopism semigroups of an equivalence relation

Zhuchok Yurii, Toichkina Olena

Luhansk Taras Shevchenko National University (Starobilsk, Ukraine)

Berlin, October 13, 2017

Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References 1. Introduction

Endomorphism semigroups of algebraic systems and their properties have been investigated by many authors (see, e.g., [1]–[4]). One of the first results related to endomorphisms of binary relations was obtained by Gluskin [5]; it states that any quasi-order relation is completely determined by the corresponding endomorphism semigroup. Later in this area of research numerous results for various classes of relations were obtained. For example, Shneperman [6] demonstrated that Gluskin’s result cannot be carried over to the class of all reflexive binary relations, whereas in [7] this result was generalized to the class of so-called dense relations, and in [8], to a certain subclass of reflexive binary relations. Similar results for certain µ-ary relations were obtained by Popov [9], who introduced the notion of an endotopism.

The notion of an endotopism is closely related to that of a correspondence, introduced by Kurosh [10] for an arbitrary universal algebra. As is known, the set of all endotopisms of a relation of any arity forms a semigroup with respect to the operation of componentwise multiplication. It turns out that the endotopism semigroup of any binary relation deﬁned on some set is the correspondence of the symmetric semigroup on the same set. The various endotopism semigroups of an arbitrary equivalence relation are the main subject of study in the present talk. The regularity of the monoid of strong endomorphisms of ﬁnite undirected graphs without multiple edges was established in [11],

ﬁnite n-uniform hypergraphs in [12], and inﬁnite undirected graphs and hypergraphs in [13]. Regularity conditions for the endomorphism semigroups of equivalences were found in [14], for the endomorphism semigroups of ordered and quasi-ordered sets in [15] and [16], and for the endomorphism monoids of countable chains in [17]. An important subclass of regular semigroups is the class of coregular semigroups. The concept of coregularity on semigroups was introduced by Bijev and Todorov [18]. Investigations in this area were continued by Chvalina and Matouskova [19] and by Dimitrova and Koppitz [20]. Here we study conditions for regularity and coregularity of various endotopism semigroups for an equivalence.

Another concept considered in this work is the notion of the endotype. Depending on the conditions imposed on the endomorphism of a symmetric binary relation, Bottcher and Knauer [21] distinguished five types of endomorphisms, which were then used to define the endotype of the relation. Later, the notion of the endotype was defined for relations of any arity [22]. This concept can be used to classify relations by their endotype with respect to endomorphisms. For example, endotypes of generalized polygons were found in [23], those of the complement of finite paths in [24], and graphs of N-prisms in [25]. Here we generalize definitions introduced in [21] for endomorphisms to the case of endotopisms of binary relations and classify all equivalence relations by their endotype with respect to endotopisms.

Zhuchok Yurii, Toichkina Olena 1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.1. Endotopisms and autotopisms

An ordered pair (ϕ, ψ) of transformations ϕ and ψ of a set X is called an endotopism [4] of ρ ⊆ X × X if the condition (x, y) ∈ ρ implies (xϕ, yψ) ∈ ρ for any x, y ∈ X . The set of all endotopisms of ρ equipped with the operation of componentwise multiplication forms a monoid denoted by Et (ρ). An ordered pair (ϕ, ψ) of permutations ϕ and ψ of a set X is called an autotopism of ρ ⊆ X × X if (x, y) ∈ ρ if and only if (xϕ, yψ) ∈ ρ for any x, y ∈ X . The set of all autotopisms of ρ endowed with the operation of componentwise multiplication forms a group denoted by At (ρ).

An endotopism (ϕ, ψ) of ρ ⊆ X × X is called a half-strong endotopism if (xϕ, yψ) ∈ ρ implies the existence of x0 ∈ xϕϕ−1, y 0 ∈ yψψ−1 such that (x0, y 0) ∈ ρ. The set of all half-strong endotopisms of ρ is denoted by HEt(ρ). An endotopism (ϕ, ψ) of ρ ⊆ X × X is called a locally strong endotopism if (xϕ, yψ) ∈ ρ implies that for any preimage x0 ∈ xϕϕ−1 there exists a preimage y 0 ∈ yψψ−1 such that (x0, y 0) ∈ ρ, and analogously for any y 0 ∈ yψψ−1. The set of all locally strong endotopisms of ρ is denoted by LEt(ρ).

An endotopism (ϕ, ψ) of ρ ⊆ X × X is called a strong endotopism if (xϕ, yψ) ∈ ρ implies (x, y) ∈ ρ for any x, y ∈ X . The set of all strong endotopisms of ρ endowed with the componentwise multiplication forms a monoid denoted by SEt (ρ). An endotopism (ϕ, ψ) of ρ ⊆ X × X is called a quasi-strong endotopism if (xϕ, yψ) ∈ ρ implies that there exists a preimage x0 ∈ xϕϕ−1 which is adjacent with respect to ρ to every preimage from yψψ−1, and analogously for suitable y 0 ∈ yψψ−1. The set of all quasi-strong endotopisms of ρ is denoted by QEt(ρ).

For any binary relation ρ on X we have a chain of inclusions: Et(ρ) ⊇ HEt(ρ) ⊇ LEt(ρ) ⊇ QEt(ρ) ⊇ SEt(ρ) ⊇ At(ρ). Let X = {a, b, c, d}, ρ = {(a, a), (a, c), (c, b), (d, a), (a, d), (d, d)}, then

a b c d a b c d , ∈ SEt(ρ)\At(ρ), a b c d d b c d

Note that HEt(ρ), LEt(ρ) and QEt(ρ) do not form semigroups in general. If an endotopism (ϕ, ψ) of ρ is such that ϕ = ψ, then we obtain the corresponding notion of an endomorphism.

Lemma. (i) A pair (τ, σ) of transformations (permutations) of X is an endotopism (autotopism) of α ∈ Eq(X ) iff for any A ∈ X /α there is B ∈ X /α such that Aτ ⊆ B, Aσ ⊆ B (Aτ = B,Aσ = B). (ii) An endotopism (τ, σ) of α ∈ Eq(X ) is a strong endotopism if and only if τ ∗ : X /α → X /α : a 7→ aτ is injective. (iii) An endotopism (τ, σ) of α ∈ Eq(X ) is a quasi-strong endotopism if and only if (τ, σ) is a strong endotopism. (iv) An endotopism (τ, σ) of α ∈ Eq(X ) is locally strong iff for all A ∈ (X /α)τ ∗ and B, C ∈ Aτ ∗−1 we have Bτ = Cτ,Bσ = Cσ. (v) An endotopism (τ, σ) of α ∈ Eq(X ) is half-strong iff for any ∗ S A ∈ (X /α)τ , (A ∩ X τ) × (A ∩ X σ) = Y ∈Aτ ∗−1 (Y τ × Y σ).

Zhuchok Yurii, Toichkina Olena Lemma. The set LEt(α) of all locally strong endotopisms of α ∈ Eq(X ) forms a correspondence of End(α) if and only if one of the following conditions is satisﬁed: (i) α is the identity relation; (ii) there is a unique class A ∈ X /α such that |A| ≥ 2. Lemma. The set HEt(α) of all half-strong endotopisms of α ∈ Eq(X ) forms a correspondence of End(α) iﬀ α is trivial.

1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.6. Correspondences of End(α)

Corollary. For any equivalence α on X , semigroups Et(α), SEt(α), QEt(α) and At(α) are correspondences of End(α).

Zhuchok Yurii, Toichkina Olena Lemma. The set HEt(α) of all half-strong endotopisms of α ∈ Eq(X ) forms a correspondence of End(α) iﬀ α is trivial.

Corollary. For any equivalence α on X , semigroups Et(α), SEt(α), QEt(α) and At(α) are correspondences of End(α). Lemma. The set LEt(α) of all locally strong endotopisms of α ∈ Eq(X ) forms a correspondence of End(α) if and only if one of the following conditions is satisﬁed: (i) α is the identity relation; (ii) there is a unique class A ∈ X /α such that |A| ≥ 2.

Zhuchok Yurii, Toichkina Olena Theorem. For any equivalence α on a set X , we have  0, |X | = 1,  4, 2 ≤ |X | < ∞, α = i ,  X Endotype(X , α) = 16, 2 ≤ |X |, α = ωX ,  20, |X | = ∞, α = iX ,  23, α 6= iX , α 6= ωX .

1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References 3. Endotypes of equivalence relations

With the sequence Et(ρ) ⊇ HEt(ρ) ⊇ LEt(ρ) ⊇ QEt(ρ) ⊇ SEt(ρ) ⊇ At(ρ) we associate the sequence (s1, s2, s3, s4, s5), where si ∈ {0, 1}, i ∈ {1, ..., 5}. Here 1 stands 6= and 0 stands = at the respective position in the above sequence. The integer P5 i−1 Endotype(X , ρ) = i=1 si 2 is called the endotype [21] of ρ. Theorem. For any equivalence α on a set X , we have  0, |X | = 1,  4, 2 ≤ |X | < ∞, α = i ,  X Endotype(X , α) = 16, 2 ≤ |X |, α = ωX ,  20, |X | = ∞, α = iX ,  23, α 6= iX , α 6= ωX . Zhuchok Yurii, Toichkina Olena Theorem. (i) The correspondence Et(α), α ∈ Eq(X ), is regular if and only if α is trivial. (ii) The correspondence HEt(α), where α is a trivial equivalence relation, is regular. 1 (iii) The correspondence LEt(α), where α ∈ Eq (X ) or α = iX , is regular. (iv) The correspondence SEt(α), α ∈ Eq(X ), is regular if and only if the quotient X /α is ﬁnite.

1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4.1. Regularity 4. Regularity and coregularity of correspondences 4.2. Coregularity 5. Representations of correspondences 6. References 4.1. Regularity

Recall that a semigroup S is regular if for any a ∈ S there is x ∈ S such that axa = a. The set of all equivalence relations on X with n classes of cardinality ≥ 2 is denoted by Eqn(X ).

Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4.1. Regularity 4. Regularity and coregularity of correspondences 4.2. Coregularity 5. Representations of correspondences 6. References 4.1. Regularity

Recall that a semigroup S is regular if for any a ∈ S there is x ∈ S such that axa = a. The set of all equivalence relations on X with n classes of cardinality ≥ 2 is denoted by Eqn(X ). Theorem. (i) The correspondence Et(α), α ∈ Eq(X ), is regular if and only if α is trivial. (ii) The correspondence HEt(α), where α is a trivial equivalence relation, is regular. 1 (iii) The correspondence LEt(α), where α ∈ Eq (X ) or α = iX , is regular. (iv) The correspondence SEt(α), α ∈ Eq(X ), is regular if and only if the quotient X /α is ﬁnite.

The concept of coregularity on semigroups was introduced by G.Bijev and K.Todorov [6] in 1980. A semigroup S is coregular if for any a ∈ S there is b ∈ S such that

aba = bab = a.

The element b is called coinverse for a. It is known that a semigroup S is coregular if and only if a3 = a for any a ∈ S or S is a union of disjoint groups at which the elements are of an order ≤ 2.

Theorem. (i) The correspondence Et (α) , α ∈ Eq(X ), is coregular if and only if |X | ∈ {1, 2}. (ii) The correspondence HEt(α), where α is the trivial equivalence relation, is coregular if and only if Et (α) is coregular. (iii) The correspondence SEt(α), α ∈ Eq(X ), is coregular if and only if |X | ∈ {1, 2} or |X | = 3, α∈ / {iX , ωX }. 1 (iv) The correspondence LEt(α), where α ∈ Eq (X ) or α = iX , is coregular if and only if SEt (α) is coregular. (v) The correspondence At (α) , α ∈ Eq(X ), is coregular if and only if SEt (α) is coregular or |X | = 4, |X /α| = 3.

Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 5.1. Representations of the monoid Et (α) 3. Endotypes of equivalence relations 5.2. Representations of the monoid SEt (α) 4. Regularity and coregularity of correspondences 5.3. Representations of the autotopism group At (α) 5. Representations of correspondences 6. References 5.1.1. n-fold wreath product of a monoid and a category

Let C be a small category, R be a monoid that acts on the right on S X = ObC, and M = MorC(x, y). Take n ∈ N and let x,y∈X

(1) (n) (i) (i) V = {(r, f , ..., f )| r ∈ R, f ∈ Map(X , M), xf ∈ MorC(x, xr)∀x},

(1) (n) (1) (n) (1) (1) (n) (n) (r, f , ..., f )(p, g , ..., g ) = (rp, f gr , ..., f gr ),

(i) (i) (i) (i) where x(f gr ) = xf (xr)g for all x ∈ X . The monoid V is called the n-fold wreath product of the monoid R with the category C and is denoted by R wr (n) C.

It should be noted that for n = 1 we obtain a dual construction to the wreath product deﬁned by V.Fleisher [26]. Theorem. Let α be an equivalence relation on a set X and let K be the small category whose objects are equivalence classes of X /α and morphisms are mappings. Then the semigroups Et(α) and = (X /α) wr (2)K are isomorphic. Corollary. For every equivalence α on a ﬁnite set X X Y |Et(α)| = ( |xϕ||x|)2. ϕ∈=(X /α) x∈X /α

The structure of isomorphisms between endotopism semigroups of arbitrary equivalence relations describes the following theorem. Theorem. Let α be an equivalence relation on a set X , and let β be an equivalence relation on a set X 0. A mapping Υ of the correspondence Et (α) onto the correspondence Et (β) is an isomorphism if and only if it can be represented in the form

gΥ = τ −1gτ for all g ∈ Et (α) ,

where τ is an isotopism of (X , α) onto (X 0, β).

We say that binary relations are determined by their endotopism semigroups in a class Θ if for all α, β ∈ Θ the condition Etα =∼ Etβ implies that α =∼ β. Theorem. Let α be an equivalence relation on X , and let β be an equivalence relation on Y . If the correspondences Et (X , α) and Et (Y , β) are isomorphic, then the relational systems (X , α) and (Y , β) are isomorphic.

Denote by K 2 the product of the category K (deﬁned above) by 2 2 itself and by K∗ the complete subcategory of K considered on the 2 set of objects ObK∗ = {(A; A)|A ∈ ObK}.

Let α be an equivalence relation on a set X . Denote by =in(X /α) the semigroup of all injective transformations of the quotient set X /α and single out the following subsemigroup in the Cartesian square of the wreath product =in(X /α)wrK:

α SX = {((ϕ, f ) , (ψ, g)) | ϕ = ψ} .

Zhuchok Yurii, Toichkina Olena Corollary. For every equivalence α on a ﬁnite set X X Y |SEt(α)| = ( |xϕ||x|)2. ϕ∈S(X /α) x∈X /α

1. Introduction 2. Endotopisms of an equivalence relation 5.1. Representations of the monoid Et (α) 3. Endotypes of equivalence relations 5.2. Representations of the monoid SEt (α) 4. Regularity and coregularity of correspondences 5.3. Representations of the autotopism group At (α) 5. Representations of correspondences 6. References 5.2. Representations of the monoid SEt (α)

Theorem. Let α be an equivalence relation on X . The monoid SEt(α) is isomorphic to each of the following semigroups: α (i) the subdirect product SX of the monoid (=in(X /α))wrK) × (=in(X /α))wrK); 2 (ii) the wreath product =in(X /α)wrK∗ of the monoid =in(X /α) 2 with the category K∗ ; (2) (iii) the 2-fold wreath product =in(X /α)wr K of the monoid of transformations =in(X /α) with the small category K.

Let us introduce a small category K by setting Ob K = X /α, Mor a, b = Mapb a, b , where Mapb a, b is the set of all bijections from the class a to the class b. Denote by B(X /α) the set of all bijections δ : X /α → X /α such that |a| = |aδ| for every a ∈ X /α. It is clear that B(X /α) is a subgroup of the group of all permutations on the quotient set X /α. Take the following subgroup of the direct product of B (X /α) wrK by itself: α AX = {((ϕ, f ) , (ψ, g)) | ϕ = ψ} .

Theorem. Let α be an equivalence relation on the set X . The group At(α) is isomorphic to each of the following groups: α (i) the subdirect product AX of the group (B (X /α) wrK) × (B (X /α) wrK); 2 (ii) the wreath product B (X /α) wrK∗ of the permutation group 2 B (X /α) with the category K∗; (2) (iii) the 2-fold wreath product B (X /α) wr K of the permutation group B (X /α) with the small category K. Corollary. For every equivalence α on a ﬁnite set X Y |At (α)| = |B(X /α)| · ( |A|!)2. A∈X /α

Denote by P the set of all representatives taken precisely once in every equivalence class of the cardinality relation ∼ on X /α, and let Y = {i|Ki ∈ P}. For every i ∈ Y , put [ i = {j ∈ J | Ki ∼ Kj }, Xi = Kj and αi = α ∩ (Xi × Xi ). j∈i Theorem. The autotopism group At (α) , α ∈ Eq(X ), is Q isomorphic to the direct product i∈Y S(Ki ) × S(Ki )WrS(i) of wreath products of the Cartesian square S(Ki ) × S(Ki ) with the symmetric group S(i). Moreover, if X is ﬁnite, then

Y 2 |At (α)| = (|Ki |! · | i |). i∈Y

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Zhuchok Yurii, Toichkina Olena