CATEGORICAL DUALITY AND CROSS

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In the following P always denotes a partially ordered set and P(x) the principal order ideal generated by x. For any two partially ordered sets P and Q, an order.
CATEGORICAL DUALITY AND CROSS-CONNECTIONS P. G. ROMEO

Abstract. The cross-connections of regular partally ordered sets introduced by P.A.Grillet and K.S.S.Nambooripad in late 1970’s to describe the fundamental regular semigroups. More recently in early 1990’s Nambooripad generalised the cross connection theory by considering the cross-connections of normal categories and thus described the structure of regular semigroups. Here we further generalze this theory by replcing the normal category by its more general analogue " the proper category" and discuss their cross-connections.

1. Prelimanires In the following P always denotes a partially ordered set and P (x) the principal order ideal generated by x. For any two partially ordered sets P and Q, an order preserving mapping f : P → Q is called normal if im f = Q(a) for some a ∈ Q and for every x ∈ P there exists an z ≤ x such that f | P (z) is an isomorphism of P (z) onto Q(xf ). Let f : P → Q be a normal map then M (f ) denotes the set of all elements b ∈ P such that P (b) is isomorphic to im f . The set of all normal mappings S(P ) on a partially ordered set P is a semigroup. The idempotents of S(P ) are called normal retractions and a principal ideal P (a) is called a normal retract if it is an image of some idempotent. Definition 1. A partially ordered set P such that every principal ideal of P is a normal retract then P a regular partially orderd set. An equivqlence ρ on the poset P is called normal if there exists a normal mapping f ∈ S(P ) such that ker f = f f −1 = ρ, the post (under reverse inclusion) of all normal equivalences on P will be denoted by P ◦ Inspired by Munn’s work fundamental inverse semigroup, P.A.Grillet introduced cross-connections of regular partially ordered sets to describe the fundamental regular semigroups [2] . Definition 2. A cross-connection between two partially ordered sets I and Λ is a pair of mapping Γ : Λ → I ∗ and ∆ : I → Λ∗ where I ∗ , Λ∗ denotes the set of all normal equivalences on I, Λ ordered by reverse inclusion. Note that the sets I ∗ [Λ∗ ] above may be regrded as duals of I [Λ] respectively. Given a cross-connection (Γ, ∆) between regular posets I and Λ then there exists a regular subsemigroup of N (I)×N (Λ)◦ where N (I) is the set of all normal mappings on I and N (Λ)◦ is the left right dual of N (Λ). This regular subsemigroup is called the cross-connection semigroup and is denoted by T[I,Λ,Γ,∆] . A regular semigroup S is called fundamental if the identity congruence is the only idempotent seperating congruence in S. 2010 Mathematics Subject Classification. 20M10. Key words and phrases. Proper category, Cones, cross-connections. 1

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Theorem 1. (Grillet’s Theorem, [2]) Let S be a fundamental regular semigroup. Them I = S/R and Λ = S/L are regulr partially ordered sets, S induces a crossconnection (Γ, ∆) between I and Λ, and S is isomorphic to a full regular subsemigroup of T[I,Λ,Γ,∆] . K.S.S. Nambooripad refined Grillet’s cross-connections of partially ordered sets and obtaind the fundamental regular semigroup which is the exact analog of Munn’s construction of fundamental inverse semigroup (see Theorem 4.9 (Nambooripad), Ch. VIII [2]). 1.1. Cross-connections of Normal categories. More recently in 1995, [5], Nambooripad further generalised the idea of cross-connections, to obtain a more general theory . In this grant generalization he replaced the regular partially ordered set by certain categories, which he call normal categories and the cross-connections as a pair of mappings (local isomorphisms) between such categories. In the following we briefly recall Nambooripad’s construction of cross-connection nromal categories from [5]. Here we assume familiarity with the definitions and basic results as given in [3]. In cross-connection theory, the categorical duality has a majore role and hence we recall here some basic ideas regarding duality. The opposite category C op of the category C is the category with vC op = vC and morphisms are reversed. The dual of a category is something analogous to the opposite category. But note that all duals are not opposits. The principle of duality states that given a true statement, the statement obtained by changing each concept by its dual and the order of composution is also a true statement. Example 1. Given categories C and D then [C; D] is the category whose objects are functors from C to D and morphisms are natural transformations. Any subcategory of [C; D] will be called a functor category from C to D. The category C ∗ denote the functor category [C; Set]. Obviously C ∗ may be regarded as a dual of C. Definition 3. Let C and D be two categories for d ∈ vD and F : C → D is a functor. By a cone σ we men a natural transformation σ belongs to either N at [F, ∆ d] or N at [∆ d, F ]. If σ ∈ N at [F, ∆ d] then it is called a cone from the base F to the vertex d and if σ ∈ N at [∆ d, F ] it is called a cone to the base F from the vertx d. In the following the category C is always a category with subobjects and factorization (see cf.[5]). Definition 4. For d ∈ vC. A map γ : vC → C is a cone from the base vC to the vertex d if γ satisfies the following: (1) γ(c) ∈ C(c, d) for all c ∈ vC. (2) If c0 ⊆ c then jcc0 γ(c) = γ(c0 ). A cones γ ∈ C such that there exists at least one c ∈ vC with γ(c) : c → cγ is an isomorphism is called a normal cone and the set of all normal cones in C will be denoted as T C. Definition 5. A category C with subobjects and factorization is a normal category if for each c ∈ C there exists a γ ∈ T C such that c = cγ . The set of all normal cones T C in the category C is a semigroup with respect to the binary operation defined γ · η = γ ? η(cγ )o

CATEGORICAL DUALITY AND CROSS-CONNECTIONS

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where γ, η ∈ T C. The set M γ = {c ∈ vC : γ(c) is an isomorphism } known as the M -set of γ. Given a normal category C, its dual denoted by N ∗ C, is a full subcategory of C ∗ with vN ∗ C = {(, −) :  ∈ E(T C)}. Let C and D be normal catgories. A local isomorphism Γ : C → N ∗ D is called a connection of C with D and the unique local isomorphism Γ∗ : C 0 → N ∗ D where C 0 is an appropriate ideal of C such that c ∈ Γ(d) ⇔ d ∈ Γ∗ (c) is the dual connecton. Note that the connections and dual connections together determines a bifunctor χ : C × D → Set and this natural isomorphism is often termed as the cross-connection. Theorem 2. (Nambooripad’s Theorem) Let Γ : D → N ∗ C be a connection of normal categories D with CΓ where CΓ is an (approriate) ideal of C. Then there exists a unique (dual) connection Γ∗ : CΓ → N ∗ D such that c ∈ Γ(d) ⇔ d ∈ Γ∗ (c) and χΓ : Γ(−, −) → Γ∗ (−, −) is a natural isomorphism. 2. Crossconnections of Proper Categories In the following we further generalize the cross-connection theory to include more general class of semigroups. C be a category with subobjects and factoriztion, a cones γ such that there exists at least one c ∈ vC with γ(c) : c → cγ is epimorphism is called a proper cone and the set of all proper cones in C will be denoted as PC. Let γ be a proper cone in C and f ∈ C(cγ , c). Then γ ? f o is a proper cone with vertex Imf . The component of γ ? f o at c is (γ ? f o )(c) = γ(c) ◦ f o

for all

c ∈ vC

and for all composable pair of morphisms f, g ∈ C with domf = cγ then γ ? (f g)o = (γ ? f o ) ? (g|Imf )o . Proposition 1. The set of all proper cones PC in the category C is a semigroup with respect to the binary operation defined γ · η = γ ? η(cγ )o where γ, η ∈ P C. The set M γ = {c ∈ vC : γ(c) is epimorphism } known as the M -set of γ. A proper cone γ ∈ PC is an idempotent cone if and only if γ(cγ ) = Icγ . Definition 6. A small category C with subobjects is called proper category if it satisfies the following: (1) every inclusion in C splits, (2) every morphism f ∈ C has unique canonical factorization and (3) for each a ∈ vC, there exists γ ∈ PC such that γ(a) = Ia . Lemma 1. Let PC be a proper ctegory, γ, γ 0 ∈ PC and  ∈ E(PC) (1)  · γ = γ if and only if there exists a unique epimorphism f : c → cγ such that γ =  ? f and c ∈ M γ. (2) γRγ 0 then M γ = M γ 0 .

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(3) if Lγ ≤ Lγ 0 then cγ ⊆ cγ 0 . The converse holds if γ, γ 0 are normal cones. Let C be a proper category. For each γ ∈ PC, define H(γ; −) : C → Set by H(γ; c) = {γ ? f o : f ∈ C(cγ , c)} H(γ; g) : γ ? f o → γ ? f g o where g ∈ C(c, c0 ) then H(γ; −) is called the hfunctor. Proposition 2. For γ, γ 0 ∈ PC, we have the following. (1) H(γ; −) ⊆ H(γ 0 ; −) if and only if there exists a unique epimorphism h from cγ to cγ 0 such that γ = γ 0 ? h. (2) If Rγ ≤ Rγ 0 then H(γ; −) ⊆ H(γ 0 ; −). Definition 7. If C is a proper category, then the proper dual of C denoted by P? C is the full subcategory of C ? with vP ? C = {H(γ; −) : γ ∈ PC} and morphisms are (appropriate) natural transformtions σ : H(γ; −) → H(γ 0 ; −). In particular σ is the inclusion H(γ; −) ⊆ H(γ 0 ; −) if and only if γ = γ0 ? σ ˆ. Moreover the map σ → σ ˆ is a bijection of P? C(H(γ; −), H(γ 0 ; −)) onto C(cγ , cγ 0 ). Lemma 2. If f : c → d is an epimorphism in C, then H(γ; f ) : H(γ; c) → H(γ; d) in P ? C is an epimorphism. Dually if f : c → d is a monomorphism in C then H(γ; f ) : H(γ; c) → H(γ; d) in P ? C is also a monomorphism. Note that for any proper category C the dual category P ? C is also a proper category. Given two proper catgories C and D, we call a local isomorphism Γ : C → N ∗ D a connection of C with D. Theorem 3. Let Γ : D → N ∗ C be a connection of proper categories and CΓ is an (approriate) ideal of C. Then there exists a unique (dual) connection Γ∗ : CΓ → N ∗ D such that c ∈ Γ(d) ⇔ d ∈ Γ∗ (c) and χΓ : Γ(−, −) → Γ∗ (−, −) is a natural isomorphism. References [1] A.H.Clifford and G.B.Preston(1961) The algebraic theory of semigroups vol 1 , Math Surveys No.7, American Mathematical Society, Providence,R.I. [2] P.A.Grillet(1995) Semigroups : An Introduction to Structure Theory, Marcel Dekker, INc. ISBN 0-8247-9662-4. [3] S.Mac Lane(1971) Categories for the working mathematician, Springer Verlag, Newyork, ISBN 0-387-98403-8 [4] K.S.S.Nambooripad and F.J.Pastijn (1985), Fundmental representation of a strongly regular Baer semigroup, Journal of Algebra, Vol.92, No. 2, 283-302. [5] K.S.S.Nambooripad (1994) Theory of cross connections, Publication No.28 - Centre for Mathematical Sciences, Trivandrum. Professor, Dept. of Mathematics, Cochin University of Science and Technology, Kochi, Kerala, INDIA. E-mail address: romeo− [email protected]