Map Composition Generalized to Coherent Collections of Maps

Front. Math. China 2015, 10(3): 547–565 DOI 10.1007/s11464-015-0435-5

Map composition generalized to coherent collections of maps

Herng Yi CHENG1, Kang Hao CHEONG1,2

1 National University of Singapore High School of Mathematics and Science, Singapore 129957, Singapore 2 Tanglin Secondary School, Singapore 127391, Singapore

c Higher Education Press and Springer-Verlag Berlin Heidelberg 2015

Abstract Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is deﬁned as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set∗, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements. Keywords Relation algebra, partial algebra, composition MSC 08A02

1 Introduction

This paper considers partial algebras on maps (similar to those deﬁned in [15]), constructed in analogy to relation algebras [5,12,16]. The sum of two maps f0 : X0 → Y0 and f1 : X1 → Y1 is deﬁned if the union of their graphs is a functional binary relation

R ⊆ (X0 ∪ X1) × (Y0 ∪ Y1), in which case the sum is the map (f0 + f1): X0 ∪ X1 → Y0 ∪ Y1 with graph R. This is deﬁned similarly to the union operation + from [10]. Map composition corresponds to relation composition from relation algebra. These deﬁnitions give rise to the following properties, some familiar from relation algebra.

Received October 9, 2013; accepted September 16, 2014 Corresponding author: Kang Hao CHEONG, E-mail: [email protected] 548 Herng Yi CHENG, Kang Hao CHEONG

• Map composition is associative when deﬁned. • The identity maps serve as map composition identities. • Map summation is idempotent, commutative, and associative. • Map composition right-distributes over map summation. • The empty map is a summation identity. We generalize this partial algebra, while preserving the above properties, by augmenting any map h: X → Y with additional structure, that is, associating each x ∈ X with a set of maps H ⊆ Y X such that

h(x)=h(x), ∀ h ∈ H. (1.1)

Such a structure is called a polymap (Definition 2.1). Polymaps and the operations between them are shown to be significant extensions of maps, and their operations, that exhibit several useful properties. Thus, we have algebras that preserve many of their properties when we augment their elements, maps, with collections of other elements. This is a potentially novel insight with regards to algebra. Polymaps are motivated from an abstract generalization of the technique of grafting in origami as applied to substrates P such as topological surfaces and even discrete substrates like abstract graphs. Origami is generally defined as a continuous path isometry f : P → R3 that does not cause self-intersection, where P is the initial sheet of paper. Grafting is an established origami design 3 technique, which involves folding a sheet P0 via an origami f0 : P0 → R to some configuration f0(P0), after which f0(P0) is treated as a fresh sheet of paper P1 3 and another origami f1 : P1 → R is folded on it. This process is a potent origami design technique [3] using only one sheet P0. This can be stated more precisely by considering Belcastro and Hull’s definition of origami as a piecewise rigid map [1]. That is, each origami f : P → 3 R corresponds to a family {(Pi,Ai)}i∈I such that {Pi}i∈I is a partition of P, R3 → R3 | | and each Ai : is a proper rigid transformation such that f Pi = Ai Pi for all i ∈ I (corresponding to (1.1)). As a polymap, each x ∈ Pi is assigned the set {Ai}. The result of grafting an 3 origami g : f(P ) → R (corresponding to {(Qj,Bj)}j∈J )afterf is an origami 3 P → R , for which every x ∈ Pi is assigned the set {Bj ◦ Ai} if f(x) ∈ Qj. This forms the basis of Definition 2.6 below for polymap composition. Section 2 defines polymaps and develops the basic properties of composition on polymaps, as well as inverse polymaps, in analogy to maps. Section 3 generalizes the category Set to the categories SetP, SetTP,andSetM,whose objects are sets and hom-sets consist of polymaps. Definitions from category theory follow [11] and [17]. Group-like algebraic structures and semilattices of polymaps are constructed based on the polymap sum in Section 4, including an idempotent involution semiring that has no nontrivial natural analogue in terms of maps. Polymaps are also represented as small subcategories of the category of pointed sets, Set∗, in Section 5. Map composition generalized to collections of maps 549

2 Polymaps and their operations

Deﬁnition 2.1 (Polymap) For any sets X and Y and a map f : X → Y, where X ⊆ X, a polymap from X via f to Y is a triplet f =(X, φ, Y ), where φ: X → P(Y X )isamapsuchthatφ(x) = ∅⇐⇒x ∈ X and f(x)=f (x)for all f ∈ φ(x) (corresponding to (1.1)). Such a polymap is denoted as f : X Y. Let dom f = X, f = φ, and im f = f (X). Clearly, f is unique to f so it ← is denoted as f . If |φ(x)| 1 for all x ∈ X, then f is called a monomap.If X = X, then f is called total. Example 2.2 Given any nonempty set of maps F ⊆ Y X , the constant polymap via F is deﬁned as CF : X Y, where F, x ∈ Eq F, CF (x)= ∅,x/∈ Eq F, and Eq F = {x ∈ X |∀f0,f1 ∈ F : f0(x)=f1(x)} is the equalizer of F.

Define a partial order on the class of all polymaps, where f0 f1 (f0 is a subpolymap of f1) if and only if dom f0 ⊆ dom f1 and f0 (x) ⊆ f1 (x)for all x ∈ dom f. If f0 f1, and dom f0 =domf1 (resp. f0 (x)= f1 (x) for all x ∈ dom f0), call f0 a wide (resp. full) subpolymap of f1. This nomenclature is motivated by the interpretation of polymaps as categories in Section 5, in particular Lemma 5.2. Example 2.3 Givenanysingletonset{y}, the only total polymap from a given set X to {y} is the total monomap C{f}, where f : X →{y} is a constant map. The polymaps f : X {y} are precisely the subpolymaps of C{f}. Example 2.4 Given some X ⊆ X, the inclusion polymap on X from X is X defined as 1X : X X, where {id},x∈ X, X 1X (x)= ∅,x/∈ X. ←− X The term “inclusion polymap” is motivated by the observation that 1X is the inclusion map X → X. Example 2.5 The only polymap from ∅ to any given set Y is the polymap (∅,p,Y), where p: ∅→P(Y X ) is an empty map. Such a polymap is vacuously a total monomap. 2.1 Polymap composition A composition operation on polymaps is defined in direct analogy to map composition. Let f : X Y, g: Y Z, and h: Z W be polymaps. Given any F ⊆ Y X and G ⊆ ZY , let G F = {g ◦ f | g ∈ G, f ∈ F }. Note that is associative and F ∅ = ∅ = ∅ F. 550 Herng Yi CHENG, Kang Hao CHEONG

Definition 2.6 Define the polymap composite of g after f as g • f : X Z, where ⎧ ⎨ ← g ( f (x)) f (x),x∈ dom f, • g f (x)=⎩ ∅,x∈/dom f. It can be verified easily that every polymap composite is a polymap. The analogy with map composition is shown in the following lemma. ←−1 ←−− ← ← Lemma 2.7 Each x ∈ dom (g • f)= f (dom g) satisfies g • f(x)=g ( f (x)). Proof From the definition of polymap composition, dom (g • f) ⊆ dom f. Each x ∈ dom f satisfies f (x) = ∅ and

x ∈ dom (g • f) ⇐⇒ g • f (x) = ∅ ← ⇐⇒ g ( f (x)) = ∅ ← ⇐⇒ f (x) ∈ dom g ←−1 ⇐⇒ x ∈ f (dom g).

←−1 ← Suppose that x ∈ f (dom g). Choose any f ∈ f (x)andg ∈ g ( f (x)); by the deﬁnition of polymap composition, g ◦ f ∈ g • f (x). By the deﬁnition of a polymap, ←−− ← ← g • f(x)=g(f(x)) = g ( f (x)). Lemma 2.8 Polymap composition is associative. That is,

h • (g • f)=(h • g) • f.

Proof It is ﬁrst demonstrated that

←−−−1 ←−1 ←−1 g • f (dom h)= f (g (dom h)).

←−1 ←−1 ←−1 (⊇) For all x0 ∈ f (g (dom h)) ⊆ f (dom g), Lemma 2.7 gives

←−− ← ← g • f(x0)=g ( f (x0)) ∈ dom h. ←−−−1 (⊆)Eachx1 ∈ g • f (dom h) ⊆ dom (g • h)satisﬁes g • f (x1) = ∅, and ←−1 thus, x1 ∈ f (dom g). By Lemma 2.7,

← ← ←−− g ( f (x1)) = g • f(x1) ∈ dom h.

Thus,

←−1 ←−1 dom (h • (g • f)) = f (g (dom h)) = dom ((h • g) • f). Map composition generalized to collections of maps 551

←−1 ←−1 For any x ∈ f (g (dom h)), we have ←−− h • (g • f) (x)= h (g • f(x)) g • f (x) ← ← ← (by Lemma 2.7) = h ( g ( f (x))) g ( f (x)) f (x) ← = h • g ( f (x)) f (x) = (h • g) • f (x). Remark 2.9 Given polymaps f : X Y and g: Y Z, it may be possible to define a generalized polymap composite h: X Z such that h (x)hasa ←−1 useful nonempty definition even if x/∈ f (dom g), as opposed to Definition ← 2.6. This can be achieved if f (x) “derives” its transformations under g from its surroundings. That is, if C is the component of im f \ dom g containing ← f (x), and g has constant value G over the boundary of C, then h (x)may ← be defined as G f (x) instead of g ( f (x)) f (x). However, it is likely that a topological context is unnecessary, thus the bare notion of “connectedness” may be utilized, as described using the connective spaces of Muscat and Buhagiar [13] and connectivity spaces of Dugowson [7] (or the objects of Börger’s category Zus [2]).

2.2 Inverse polymap f is called polyinjective (resp. polysurjective )ifeverymemberof x∈X f (x)is injective (resp. surjective). If f is both polyinjective and polysurjective, then it ← is called polybijective.Iff is polybijective and f is injective, then it is called invertible,inwhichcaseitsinverse is defined as f−1 : Y X, where ⎧ ← ⎨ −1 {f | f ∈ f (x)}, ∃ x ∈ X : f (x)=y, −1 f (y)=⎩ ← ∅, ∃ x ∈ X : f (x)=y. Inverting a polymap (finding its inverse) is clearly an involutory operation; the inverse polymap is clearly invertible. Other properties of this involution, especially in relation to semigroup operations [4,8], are developed in Subsection 4.2. The polymap inverse is not entirely an analogue of map inverse (consider Theorem 3.5); instead, its properties are more closely related to those of relation inverse. For example, the following lemma corresponds to the switching of relation image and preimage under the relation inverse. Lemma 2.10 If f is invertible, then dom (f−1)=imf. ←− Lemma 2.11 If f is invertible, then the graph of f−1 is the relation inverse ← of the graph of f . ←− −1 −1 0,x0 , e ∈ y0 . Proof←−For each (y ) in the graph of f pick some f ( ) We −1 have f (y0)=e(y0)=x0. The definition of inverse polymap guarantees the 552 Herng Yi CHENG, Kang Hao CHEONG

← −1 existence of some x0 ∈ X such that f (x0)=y0 and e ∈ f (x0). That is, ← −1 e (x0)=y0, and thus, x0 = e(y0)=x0. Therefore, f (x0)=y0 and (x0,y0) ← lies in the graph of f . ← Conversely, for each (x1,y1) in the graph of f , pick some f ∈ f (x1). We ← have f (x1)=f(x1)=y1. The definition of inverse polymap guarantees that ←− ←− −1 −1 −1 −1 −1 f ∈ f (y1), that is, f (y1)=f (y1)=x1. Therefore, the graph of f contains (y1,x1). ←− ← To conclude, the graph of f−1 is the relation inverse of the graph of f . Lemma 2.12 Suppose that f is invertible. Then the following statements are equivalent : (i) f−1 is total, ← (ii) f is surjective, ←− ←−1 (iii) f−1 = f . Proof (i) ⇐⇒ (ii) By Lemma 2.10, dom f−1 = Y if and only if the image of ← f is im f = Y. ← ⇒ (ii) = (iii)←− If f is surjective, then it is also bijective. Lemma 2.11 guarantees that f−1 is bijective as well because it is the relation inverse of the ← ←− ←−1 graph of f . Together these imply that f−1 = f . ← (iii) =⇒ (ii) If f is invertible, it must be surjective. Remark 2.13 The similarities between polymaps and relations described above, the origins of polymaps from relation algebra, as well as the fact that polymaps need not be total (corresponding to left-totality of relations), suggest a possible generalization of polymaps to polyrelations. A polyrelation could be defined as a relation R ⊆ X × Y with every (x, y) ∈ R being assigned a relation S ⊆ X × Y containing (x, y). Polymaps are effectively polyrelations in which R and every S are functional.

3 Categories of polymaps

The properties of polymaps and their composition allow the formation of a category SetP (resp. SetM, SetTP) on the class of sets, whose morphisms from X to Y are polymaps (resp. monomaps, total polymaps) from X to Y. The morphisms are composed via polymap composition, and the identity on X X SetM SetTP SetP is 1X . and are wide subcategories of . There is a full functor S : SetTP → Set that maps sets to themselves and every total polymap f to ← f . Consider the map P : Set → SetP which maps sets to themselves and Map composition generalized to collections of maps 553 every morphism f to C{f}. The functor P with codomain restricted to SetP (resp. SetM, SetTP)isanembedding [11] from Set to SetP (resp. SetM, SetTP). Deﬁne an equivalence relation ≈ on polymaps, where given f0 : X0 Y0 and ← ← f1 : X1 Y1, f0 ≈ f1 if and only if (X0,Y0)=(X1,Y1)andf0 = f1. Recall the deﬁnition of a congruence, which allows an equivalence relation on morphisms to induce a quotient category [11]. Lemma 3.1 ≈ is a congruence on any subcategory of SetP. Proof Consider any subcategory C of SetP,aswellassome

f0 ≈ f1 ∈ homC(X, Y ), g0 ≈ g1 ∈ homC(Y,Z).

← ← ← ← It suﬃces to show that g0 • f0 ≈ g1 • f1. Since f0 = f1 and g0 = g1, note that ←−1 ←−1 dom (g0 • f0)=f0 (dom g0)=f1 (dom g1)=dom(g1 • f1). By Lemma 2.7, ←−−− ← ← ← ← ←−−− g0 • f0(x)=g0(f0(x)) = g1(f1(x)) = g1 • f1(x), ∀ x ∈ dom (g0 • f0). Lemma 3.2 SetTP/≈ is isomorphic to Set. In fact, the following diagram commutes: SetTPS / Set Q F SetTP/≈ (3.1) where Q is the quotient functor and F is an isomorphism that maps sets to themselves and maps morphisms f : X → Y to S−1({f}). Proof The object parts of S, P, and F map every set to itself, so the said object parts must commute. Every f ∈ homSetTP(X, Y )satisﬁes ← ← ← −1 Q(f)={f ∈ homSetTP(X, Y ) | f = f } = S ({ f })=F (S(f)). Similar to Set, ∅ is an initial object for SetP, SetM,andSetTP (see Example 2.5). Every singleton set is a terminal object for SetTP (see Example 2.3).

Proposition 3.3 Given nonempty sets X, Y and any f ∈ homSetP(X, Y ), f is a section if and only if f is total and (f0(x0)=f1(x1)=⇒ x0 = x1) for all ∈ ∈ x0,x1 X and f0,f1 x∈X f (x). Proof (=⇒) Suppose that f is a section; there exists some polymap g: Y X • X such that g f = 1X . By Lemma 2.7, X =dom(g • f) ⊆ dom f ⊆ X, 554 Herng Yi CHENG, Kang Hao CHEONG ∈ ∈ and so f must be total. Consider any x0,x1 X and f0,f1 x∈X f (x) such that f0(x0)=f1(x1). The condition that dom (g • f)=domf implies that g (f0(x0)) = ∅, so choose any g ∈ g (f0(x0)). It follows that g◦f0 =id=g◦f1, and that x0 =id(x0)=g(f0(x0)) = g(f1(x1)) = id(x1)=x1.

(⇐=) Suppose that f is total and (f0(x0)=f1(x1)= ⇒ x0 = x1)for ∈ ∈ ∈ any x0,x1 X and f0,f1 x∈X f (x). For any f x∈X f (x),fmust be injective, so let gf denote the restriction of some retraction of f to f(X). For any y ∈ Y, the family {gf } ←−1 can be summed following [10] f∈ f( f ({y})) ←−1 because for any f0,f1 ∈ f ( f ({y})) and y ∈ f0(X) ∩ f1(X), observe that f0(gf0 (y )) = y = f1(gf1 (y )), thus gf0 (y )=gf1 (y ). There exists some monomap g: Y X, where im f =domg and g (y)= {g} for some g such that g|R = gf , (3.2) ←−1 f∈ f( f ({y})) where R = f(X). ←−1 f∈ f( f ({y}))

X To verify that g • f = 1 , observe that for any x ∈ X and f ∈ f (x),gf is X ← a restriction of g, where g ( f (x)) = {g}, thus g ◦ f =id. Hence,

g • f (x)={id}. Given a set A of maps from X to Y, let A denote the map whose graph is the intersection of the graphs of the members of A. This is an extension of the intersection operator · from [10].

Proposition 3.4 Given nonempty sets X, Y and any f ∈ homSetP(X, Y ), f ← ⊆ | is a retraction if and only if there exists some X dom f such that f X is surjective and f (x) is surjective for all x ∈ X. Proof (=⇒) Suppose that there exists some polymap e: Y X such that • Y f e = 1Y ;notethat

←−1 dom e ⊆ Y =dom(f • e)= e (dom f) ⊆ dom e.

←−1 Hence, dom e = e (dom f), and thus, im e ⊆ dom f. By Lemma 2.7,

← ← ←−− ←− | ◦ • Y ( f im e) e = f e = 1Y =id, Map composition generalized to collections of maps 555

← thus, f |im e must be surjective. Consider any x ∈ im e ⊆ dom f. Choose any ←−1 f ∈ f (x),y∈ e ({x}), and e ∈ e (y); f ◦ e =id=f|im e ◦ e so e must be injective and f|im e must be surjective. For any y ∈ Y and f ∈ f (x), notice that f (e(y)) = y = f(e(y)), thus, f|im e is a restriction of f . Therefore, f|im e is a restriction of f (x), and both must be surjective. ← ⇐ ⊆ | ( =) Suppose that there exists some X dom f such that f X is surjective and f (x) is surjective for all x ∈ X. Thereexistssomemap ← ← → | ◦ | e: Y X such that ( f X ) e =id, because ( f X ) is surjective. For any x ∈ X , choose some map εx such that ( f (x)) ◦ εx =id(since f (x)is surjective). For any y ∈ Y, deﬁne the map y : Y → X such that e(y ),y= y , y(y )= εe(y)(y ),y= y .

Given the monomap e: Y X, where e (y)={y}, it is veriﬁed that ← • Y f e = 1Y . Observe that e is simply a codomain expansion of e, so ←−1 dom (f • e)= e (dom f)=e−1(dom f ∩ X)=Y.

Note that for all y,y ∈ Y and f ∈ f (e(y)), f(e(y )),y= y , f(y(y )) = f(ε ( )(y )),y= y , ⎧ e y ← ⎨ f |X (e(y )),y= y , = ⎩ ( f (e(y)))(εe(y)(y )),y= y , =id(y) = y.

Hence, for any y ∈ Y, ← f • e (y)= f ( e (y)) e (y)={f ◦ y | f ∈ f (e(y))} = {id}. The inverse polymap is not, in general, a multiplicative inverse with respect to polymap composition. Some necessary conditions for polymaps to be sections or retractions are established as follows.

Theorem 3.5 Given nonempty sets X, Y and some f ∈ homSetP(X, Y ), ← (1) if f is a section, then f is total and polyinjective, and f is injective; ← (2) if f is a retraction, then f is surjective; 556 Herng Yi CHENG, Kang Hao CHEONG

← (3) f is an isomorphism if and only if f is total, f is surjective, and any of the following equivalent statements hold: (a) f is an invertible monomap ; −1 • X (b) f is invertible and f f = 1dom f; • −1 Y (c) f is invertible and f f = 1im f. Proof (1) If f is a section, then by setting f0 = f1 in Proposition 3.3, it follows immediately that f is total and polyinjective. For any x0,x1 ∈ dom f such that ← ← f (x0)= f (x1), choose any f0 ∈ f (x0)andf1 ∈ f (x1). It can be seen that

← ← f0(x0)= f (x0)= f (x1)=f1(x1), and thus, x0 = x1. This implies that f is injective. (2) If f is a retraction, then by Proposition 3.4, there exists some X ⊆ X ← ← such that f |X is surjective. It follows that f must also be surjective. (3) (a), (b), and (c) are ﬁrst shown to be equivalent. (a) =⇒ (c) Suppose that f is a invertible monomap. For any y ∈ im f = dom (f • f−1), let f−1 (y)={g}. It follows that ←− f • f−1 (y)= f (f−1(y)) f−1 (y)={g−1} {g} = {id}.

• −1 Y Thus, f f = 1im f. −1 Y (c) =⇒ (b) Suppose that f • f = 1im . For any x ∈ dom f,f∈ f (x), and ← f g ∈ f−1 ( f (x)), note that by (c),

← {id} = f • f−1 ( f (x)) ←− ← ← = f (f−1( f (x))) f−1 ( f (x)) ← = f (x) f−1 ( f (x)) f ◦ g.

Then f ◦g =id. However, f and g are bijections because f and f−1 are invertible, so g = f −1 and g ◦ f =id. Hence,

← f−1 • f (x)= f−1 ( f (x)) f (x)={id}.

−1 X (b) =⇒ (a) Suppose that f •f = 1dom ; f is invertible. For any x ∈ dom f, ← f choose any g ∈ f−1 ( f (x)). Since f−1 must also be invertible, g is bijective. For any f ∈ f (x),

← {id} = f−1 • f (x)= f−1 ( f (x)) f (x) g ◦ f, Map composition generalized to collections of maps 557 thus, g ◦ f =idandf = g−1. Therefore, every member of f (x)isequivalent to g−1, and f must be a monomap. Next, it is proven that f is an isomorphism if and only if f is a total invertible ← monomap such that f is surjective. (⇐=) If f is a total invertible monomap, then

−1 • X X f f = 1dom f = 1X .

(=⇒)Iff is an isomorphism, then it is both a section and a retraction. ← By (1) and (2), f is total and polyinjective, and f is bijective. By Proposition ← 3.4, there exists some X ⊆ dom f such that f |X is surjective and f (x)is ← surjective for all x ∈ X. However, since f is bijective, X must be equivalent ← ← | to X or else f X would not be surjective. Hence, f is surjective. Clearly, f must be injective for any x ∈ X =domf and f ∈ f (x); moreover, f (x) is a restriction of f so f must also be surjective. Therefore, f is bijective. This leads to the invertibility of f. Choose any two bijections f0,f1 ∈ f (x); −1 −1 −1 −1 by Proposition 3.3, f0 (y)=f1 (y) for all y ∈ Y ; this means that f0 = f1 , and consequently, f0 = f1, and | f (x)| =1. This concludes that f must be a monomap.

4 Algebraic structures from polymap sum

Algebraic structures involving polymaps are constructed by considering a sum over arbitrary families of polymaps. This operation is in turn based on the restriction of the relation union + from [10] to functional binary relations. { → } A family fi : Xi Yi i∈I of maps is called summable if the union of their ⊆ × graphs is a functional binary relationR (i Xi) ( i Yi), in which the sum → of the family is denoted as the map i fi : i Xi i Yi whose graph is R. 4.1 Polymap sum

Definition 4.1 Given a family {fi : X Y }i∈I of polymaps, the family is ← called summable if {fi }i∈I is a summable family of maps. If {fi}i∈I is summable, then its sum is defined as fi = X, x → fi (x),Y , i i which can be verified to be a polymap. In the case where I = {0, 1}, the sum can also be written as f0 + f1. When well-defined, polymap summation is clearly idempotent, commuta- Y tive, and associative, with each 0X = C∅ : X Y as an additive identity. Some immediate consequences are as follows. 558 Herng Yi CHENG, Kang Hao CHEONG

Lemma 4.2 If {fi}i∈I is summable, then

←−−− ← dom fi = dom fi, fi = fi . i i i i

4.2 Groups and related structures Consider a set X. Similar to the transformation semigroups and monoid arising from subsets of homSet(X, X), semigroups and monoids can arise from subsets of homSetP(X, X), including the endomorphism semigroup homSetP(X, X) itself. By Cayley’s Theorem [14] on semigroups, every semigroup of polymaps must be isomorphic to a semigroup of maps. Analogies to the symmetric group Sym(X)onX are of greater interest, because they inherit much of its well-behaved structure. The set of all total monomaps f : X X such that

← { f }∪ f (x) ⊆ Sym(X) x∈X forms a group under polymap composition and inverse. By Theorem 3.5, this group is AutSetP(X)=AutSetTP(X)=AutSetM(X), and it is a subgroup of the group of invertible monomaps (from X to X) under polymap composition and inverse. Let f : X Y, g: Y Z, and h: Z W be polymaps. Polymap composition distributes over polymap sum.

Theorem 4.3 If {gi : Y Z}i∈I is summable, then {gi •f}i∈I and {h •gi}i∈I are summable, and gi • f = (gi • f), h • gi = (h • gi). i i i i

∈ ∈ • ∩ • Proof Consider any i0,i1 I. For any x dom (gi0 f) dom (gi1 f), Lemma ←−−− ← • ←− { } 2.7 requires that gi0 f(x)=gi0 ( f (x)). Since gi : Y Z i∈I is summable,

← ← ←−−− ←− ←− • gi0 ( f (x)) = gi1 ( f (x)) = gi1 f(x).

{ • } ∈ • ∩ • Hence, gi f i∈I is summable. Similarly, for any y dom (h gi0 ) dom (h gi1 ), ←−−− ← • ←− { } Lemma 2.7 requires that h gi0 (y)=h (gi0 (y)). Since gi : Y Z i∈I is summable, ← ← ←−−− ←− ←− • h (gi0 (y)) = h (gi1 (y)) = h gi1 (x). Map composition generalized to collections of maps 559

Hence, {h • gi}i∈I is summable. By Lemma 4.2,

←−1 dom gi • f = f dom gi i i ←−1 = f (dom gi) i = dom (gi • f) i =dom (gi • f) . i

←−1 ∈ ⊆ For any x i f (dom gi) dom f, ← gi • f (x)= gi ( f (x)) f (x) i i ← = gi ( f (x)) f (x) i ← = gi ( f (x)) f (x) i = gi • f (x) i = (gi • f) (x). (4.1) i Similarly, by Lemma 4.2, ←−−−−1 dom h • gi = gi (dom h) i i ←−1 = gi (dom h) i = dom (h • gi) i =dom (h • gi) . i

For any ←−1 y ∈ gi (dom h) ⊆ dom gi , i∈I i∈I let J = {j ∈ I | y ∈ dom gj}. 560 Herng Yi CHENG, Kang Hao CHEONG

It follows that ←−−− h • gi (y)= h gi(y) gj (y) i∈I i∈I j∈J ← = ( h (gj (y)) gj (y)) ∈ jJ = h • gj (y) ∈ jJ = (h • gi) (y). i∈I While Theorem 3.5 shows that in general the inverse polymap is not a multiplicative inverse with respect to polymap composition, polymap inversion does satisfy many properties of involutions compatible with operations on ring- like structures. Lemma 4.4 For any sets X ⊆ X and Y, it holds that

X −1 X Y −1 X (1X ) = 1X , (0X ) = 0Y . Polymap inversion also has properties analogous to antihomomorphism. Proposition 4.5 If f and g are invertible, then g•f is invertible and (g•f)−1 = f−1 • g−1. • Proof Since every element of x∈X g f (x) is the composite of two bijections, g • f is polybijective. By Lemma 2.7,

←−− ← ← g • f = g ◦ ( f |←−1 ) f (dom g) must be injective. Hence, g • f is invertible. ← ←− ←− Note that g and g−1 are injective. By Lemma 2.11, the graph of g−1 is ← the relation inverse of the graph of g so for all Y ⊆ Y and Z ⊆ Z such that ← ←− ←− −1 ← g (Y )=Z, we have g−1(Z)=Y , which means that g−1 (Y )=Z = g (Y ). By Lemmas 2.10 and 2.7,

dom ((g • f)−1)=im(g • f) (4.2) ←−− = g • f(dom (g • f)) (4.3) ← ← ←−1 = g ( f ( f (dom g))) (4.4) ← = g (im f ∩ dom g) (4.5) ←− −1 = g−1 (im f ∩ dom g) (4.6) ←− −1 = g−1 (dom (f−1)) (4.7) =dom(f−1 • g−1). (4.8) Map composition generalized to collections of maps 561 ←−− For any z ∈ im (g • f), there exists exactly one x ∈ X such that g • f(x)=z, ←−− ← ←−1 ← due to the injectivity of g • f. Similarly, f is injective so f ({ f (x)})={x}. ← ← ←−− ←− ← Note that g ( f (x)) = g • f(x)=z so g−1(z)= f (x).

(g • f)−1 (z)={e−1 | e ∈ g • f (x)} ← = {f −1 ◦ g−1 | f ∈ f (x),g∈ g ( f (x))} ← = {f −1 | f ∈ f (x)} {g−1 | g ∈ g ( f (x))} ← = f−1 ( f (x)) g−1 (z) ←− = f−1 (g−1(z)) g−1 (z) = f−1 • g−1 (z). Polymap inversion also distributes over the polymap sum. Proposition 4.6 { } Given a summable family of polymaps fi i∈I such that i fi { −1} is invertible, each fi is also invertible, the family fi i∈I is summable, and −1 −1 ( i fi) = i fi . ←−−− Proof Since i fi is invertible, i fi must be injective. It also follows that every member of fi (x)= fi (x) x∈X i i x∈X ← ←−−− ∈ | must be bijective. Naturally, for each i I, fi = i fi dom fi is injective and every member of x∈X fi (x) is bijective, and thus, fi is invertible. For any i0,i1 ∈ I and ∈ −1 ∩ −1 ∩ y dom fi0 dom fi1 =imfi0 im fi1 (Lemma 2.10), there exists exactly one x such that ←−−− ← ←

fi(x)=fi0 (x)=fi1 (x)=y. i ∈ ∈ Choose any f0 fi0 (x)andf1 fi1 (x); note that both are bijections, and f0(x)=y = f1(x). This also implies that −1 ∈ −1 −1 ∈ −1 f0 fi0 (y),f1 fi1 (y), and thus, ←− ←− −1 −1 −1 −1 fi0 (y)=f0 (y)=x = f1 (y)=fi1 (y). { −1} Hence, fi i∈I is summable. By Lemma 2.10, −1 −1 −1 dom fi =im fi = im fi = dom fi =dom fi . i i i i i 562 Herng Yi CHENG, Kang Hao CHEONG ←−−− ∈ Finally, for any y i im fi, there exists a unique x such that i fi(x )= y. −1 −1 fi (y )={(f ) | f ∈ fi (x )} i i −1 = {f | f ∈ fi (x )} i −1 = fi (y ) i −1 = fi (y ). i Given a complete -semilattice F ⊆ homSetP(X, Y ) of which every member is invertible, Proposition 4.6 guarantees that the involution f → f−1 is a complete semilattice isomorphism from F to {f−1 | f ∈ F }. As polymaps contain more information than maps, there are algebraic structures that are peculiar to polymaps and are not direct analogues of algebraic structures on maps. Given a set X, Theorem 4.3 guarantees the existence of the following structure, similar to that described in [6], especially [9], where the involution is the relation inverse. Lemma 4.7 The set ← A = {f ∈ homSetP(X, X) | f is polybijective, f is an inclusion map} forms an idempotent involution semiring ID(X) with + as addition, • as X X multiplication, polymap inverse as the involution, as well as 0X and 1X as the additive and multiplicative identities, respectively. Proof Noting that + and • preserve polybijectivity, it can be seen that (A, +) is a commutative idempotent semigroup (equivalently, a +-semilattice). By Theorem 4.3, • distributes over +, and (A, •) is a semigroup. Moreover, if f, g ∈ A, then Proposition 4.6 gives

(f + g)−1 = f−1 + g−1 and Proposition 4.5 gives

(g • f)−1 = f−1 • g−1.

Lemma 4.4 also implies that

X −1 X X −1 X (0X ) = 0X , (1X ) = 1X . ID X The poset ( (X), )isacomplete -semilattice with least element 0X and greatest element i such that i (x)=AutSet∗ (X, x), the automorphism group of (X, x)inSet∗. Map composition generalized to collections of maps 563

Remark 4.8 Many of the algebraic structures of polymaps constructed above are natural extensions of their map analogues, which are their “projections” ← via the operator f → f . This suggests a generalization of polymaps to “higher dimensions”; an n-polymap f from X to Y could be inductively deﬁned as its ← projection, the map f : X → Y, with each x ∈ X assigned a set F of (n − 1)- ← ← polymaps from X to Y such that f (x)=f (x) for all f ∈ F. A 0-polymap would be a map, and a 1-polymap would be a polymap. This might allow the construction of analogues of familiar algebraic structures for general n-polymaps. 4.3 Partial order and semilattices Given sets X and Y, the partial order restricted to homSetP(X, Y )isthat which is induced by the idempotent addition. That is,

f0 f1 ⇐⇒ f0 + f1 = f1.

Lemma 4.9 Given some f ∈ homSetP(X, Y ), deﬁne B = {f ∈ homSetP(X, Y ) | f f}. Then the poset (B,) is a complete -semilattice, called the subpolymap semilattice of f.

Proof Consider any F ⊆ B, f0, f1 ∈ F, and x ∈ dom f0 ∩ dom f1. Clearly, f0, f1 f, and so ← ← ← f0(x)= f (x)=f1(x). Hence, F is summable, and F ∈ B. Therefore, F is a complete -semilattice. Given a complete semilattice G ⊆ homSetP(Y,Z), Theorem 4.3 guarantees that the “polymap actions” (or “composition operators”) g → g•f and g → h•g are complete semilattice homomorphisms from G to the subpolymap semilattices of ( G) • f and h • ( G), respectively.

5 Polymaps as categories

A polymap f : X Y may be regarded as a category by regarding f (x)asa ← hom-set for each x ∈ X. In fact, since every member of f (x)mapsx to f (x), ← f (x) ⊆ homSet∗ ((X, x), (Y, f (x))).

This observation motivates the interpretation of polymaps as small subcategories of Set∗ as presented below. Consider a map C from the class of all polymaps to the class of small subcategories of Set∗, such that C maps every polymap f : X → Y toasmall 564 Herng Yi CHENG, Kang Hao CHEONG subcategory C (called the categorical form of f)ofSet∗, where the objects of C are Ob(C)={(X ×{0}, (x, 0)) | x ∈ X}∪{(Y ×{1}, (y,1)) | y ∈ Y } and if an (injective) operation is deﬁned on maps such that any map f : X → Y corresponds to f: X ×{0}→Y ×{1}, where f(x, 0) = (f(x), 1), then ⎧ ← ⎨{f | f ∈ f (x)}, f (x)=y, homC((X0, (x, 0)), (Y1, (y,1))) = ⎩ ← (5.1) ∅, f (x) = y,

homC((Y1, (y,1)), (X0, (x, 0))) = ∅, (5.2) { } id ,z= z , homC((Z, (z,i)), (Z, (z ,i))) = (5.3) ∅,z= z. Lemma 5.1 C(f) is a category for any polymap f. Proof By (5.3), each object A ∈ Ob(C(f)) has an identity morphism because {id} =homC(f)(A, A). Consider the morphisms

f g (X, (x, i)) −→ (Y , (y,j)) −→ (Z, (z,k)) −→h (W , (w, l)). By (5.2), 0 i j k l 1. Hence, at most one e ∈{f ,g,h} is not the identity map, in which case h ◦ (g ◦ f )=e =(h ◦ g) ◦ f . Note that monomaps are precisely the polymaps with thin categorical forms. C is clearly injective, allowing the representation of polymaps as small subcategories of Set∗, which are called categorical polymaps. Lemma 5.2 below justiﬁes the terms “subpolymap”, “wide subpolymap”, and “full subpolymap”. Recall that a subcategory is wide when it inherits all of the objects from its parent space, and full when it inherits all morphisms between every pair of inherited objects [17].

Lemma 5.2 Given polymaps f0, f1 : X Y, f0 is a subpolymap (resp. wide subpolymap, full subpolymap) of f1 if and only if C(f0) is a subcategory (resp. wide subcategory, full subcategory) of C(f1). The inverse polymap relates to the opposite category. Lemma 5.3 If f is invertible, then C(f−1) is isomorphic to C(f)op. Proof Construct the required isomorphism F : C(f)op → C(f−1) such that it maps each object (X ×{i}, (x, i)) to (X ×{1−i}, (x, 1−i)). When F is restricted to each hom-set homC(f)op ((Y , (y,j)), (X , (x, i))), set F as a constant map with ← value id if i = j. Otherwise, if i

References

1. Belcastro S, Hull T C. Modelling the folding of paper into three dimensions using affine transformations. Linear Algebra Appl, 2002, 348: 273–282 2. Börger R. Connectivity spaces and component categories. In: Categorical Topology, Proc Int Conf, Toledo/Ohio 1983. Sigma Ser Pure Math, 5. 1984, 71–89 3. Cheng H Y, Cheong K H. Designing crease patterns for polyhedra by composing right frusta. Comput -Aided Design, 2012, 44: 331–342 4. Crvenković S, Dolinka I. Varieties of involution semigroups and involution semirings: a survey. Bull Soc Math Banja Luka, 2002, 9: 7–47 5. De Morgan A. On the syllogism: IV, and the logic of relations. T Cambridge Philos Soc, 1864, 10: 331–358 6. Dolinka I. Idempotent distributive semirings with involution. Internat J Algebra Comput, 2003, 13: 597–625 7. Dugowson S. On connectivity spaces. Cah Topol Géom Différ Catég, 2010, 51: 282–315 8. Easdown D, Munn W D. On semigroups with involution. Bull Aust Math Soc, 1993, 48: 93–100 9. Esik´ Z, Bernátsky L. Equational properties of Kleene algebras of relations with conversion. Theoret Comput Sci, 1995, 137: 237–251 10. Givant S. The calculus of relations as a foundation for mathematics. J Automat Reason, 2006, 37: 277–322 11. Mac Lane S. Categories for the Working Mathematician. Berlin: Springer, 1978 12. Maddux R D. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 1991, 50: 421–455 13. Muscat J, Buhagiar D. Connective spaces. Mem Fac Sci Eng, Shimane Univ Ser B Math Sci, 2006, 39: 1–13 14. Nummela E C. Cayley’s theorem for topological groups. Amer Math Monthly, 1980, 87: 202–203 15. Reyes M L. Obstructing extensions of the functor spec to noncommutative rings. Israel J Math, 2012, 192: 667–698 16. Tarski A. On the calculus of relations. J Symbolic Logic, 1941, 6: 73–89 17. Taylor P. Practical Foundations of Mathematics. Cambridge: Cambridge University Press, 1999