Cross Connection of Boolean Lattice

Transcription

1 International Journal of Algebra, Vol. 11, 2017, no. 4, HIKARI Ltd, Cross Connection of Boolean Lattice P. G. Romeo P. R. Sreejamol Dept. of Mathematics Cochin University of Science Technology Kochi, Kerala, India Copyright c 2017 P. G. Romeo P. R. Sreejamol. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract It is known that the cross connection semigroup of a complemented modular lattices is a strongly regular Baer semigroup conversely if S is a strongly regular Baer semigroup then L l, L r the poset of principal left right ideals of S are dually isomorphic complemented modular lattices. In this paper we discuss the cross connection of complemented distributive lattice showns that the cross connection determines a Boolean ring, which is a strongly regular Baer ring. Further it is also proved that the ideals of such a Boolean ring form a complemented distributive lattice. Mathematics Subject Classification: 06E05, 06D05, 06E25, 20M10 Keywords: Boolean lattice, Boolean ring, Cross connection 1. Introduction A cross connection is a categorical duality which turns out to be very significant in the study of the structure of the objects under consideration. In [3] P.A.Grillet described the cross connection of partially ordered sets obtained the fundamental regular semigroup as cross connection semigroup. K.S.S.Nambooripad F.J.Pastijn together established the cross connection of a complemented modular lattice obtaied the strongly regular Baer semigroup in [1]. Further in [5] K.S.S.Nambooripad generalized the cross connection theory to include arbitrary regular semigroups by replacing the partially ordered sets with normal categories. In the following we describe the cross

2 172 P. G. Romeo P. R. Sreejamol connection of boolean lattice obtained its representation as a cross connection ring. 2. Preliminaries A lattice is a partially ordered set (poset) L such that every finite subset of L has a least upper bound a greatest lower bound called the join meet respectively. A lattice is distributive if : a (b c) = (a b) (a c), a (b c) = (a b) (a c), a, b, c L. If a lattice L contains a smallest (greatest) element, then it is uniquely determined is denoted by 0 (1). A lattice in which every subset has meet join is a complete lattice. Definition 1. Let L is a distributive lattice, if for every a L there exists an element b L such that a b = 0 a b = 1, then b is a complement of a. If every a L has a complement then L is a complemented distributive lattice (a Boolean lattice). Definition 2. Let A B be Boolean lattices. A (boolean) homomorphism is a mapping f : A B such that, for all p, q A: (1) f(p q) = f(p) f(q), (2) f(p q) = f(p) f(q) (3) f(a c ) = f(a) c. A boolean ring is a ring (R, +, ) with unit in which every element is a multiplicative idempotent. Example 1. Let (X, Σ) be a measurable space R = {χ A : A Σ}, where χ A denotes the characteristic function of A. Then R together with Boolean operations χ A χ B = χ A B χ A χ B = χ A B A, B Σ A B = (A B c ) (A c B) is a Boolean ring. An element a in a ring R is regular (von Neumann regular) if there exists x R such that axa = a. The ring R is regular if every a R is regular R is strongly regular if each a satisfies a = xa 2 = a 2 x for some x R. A Boolean ring is von Neumann regular strongly regular. The subset E l = {y R yx = 0}, x E} is called the left annihilator of R E r = {y R xy = 0}, x E} is the right annihilator. Definition 3. A ring R is said to be Baer if every left (equivalently, right) annihilator ideal is generated by an idempotent. A strongly regular ring R is Baer if only if the Boolean lattice B(R) of all idempotents in R is complete.

3 Cross connection of Boolean lattice 173 Definition 4. An ideal in a partially ordered set P is a subset I of P such that x y I implies x I. The principal ideal P (x) of P generated by x P is P (x) = {y P y x}. Definition 5. Let P Q be two partially ordered sets. A mapping f : P Q is said to be normal mapping if it is order preserving, imf = Q(a) for some a in Q x P, there exists z x such that f P (z) is an isomorphism from P (z) onto Q(xf). If f : P Q is normal, then there exists at least one element b in P such that f is an isomorphism of P (b) onto Q(a) = imf we denote by M(f) the set of all elements b P with this property. The set of all normal mappings from P to P, denoted by S(P ) is a semigroup under composition. Idempotent normal mappings are called normal retractions a principal ideal P (a) is called normal retract if P (a) = ime where e is some normal retraction. P is called regular poset if every principal ideal of P is a normal retract. An equivalence relation ρ on a poset P is said to be normal if there exists a normal mapping f S(P ) such that kerf = ρ. The poset (under the reverse of inclusion) of all normal equivalences on P is denote by P o P o is regular whenever P is. For each ρ P o we may associate the subset M(ρ) defined by M(ρ) = M(f), where f is any normal mapping with kerf = ρ a M(ρ) if only if P (a) intersects every ρ - class in exactly one element. Then P (a) ρ(x) contains a single element which is minimal in its ρ - class the mapping ε p (ρ, a) which sends each x in P to the unique element in P (a) ρ(x) is a normal retraction with kerε p (ρ, a) = ρ imε p (ρ, a) = P (a). ε p (ρ, a) is called the projection along ρ upon P (a). If f is a normal mapping with domf = P kerf = ρ then for any a M(f) = M(ρ), the map f is factorized as f = ε p (ρ, a)α where α = f P (a) is an isomorphism of P (a) onto imf this factorization is called a normal factorization of f. The following proposition is from [1]. Proposition 1. Let I Λ be regular partially ordered sets f : I Λ be a normal mapping. For σ Λ o, define f o (σ) = ker(fɛ Λ (σ, u)) = σf 1 where u M(σ). Then f o : Λ o I o is a normal mapping such that imf o = I o (kerf) M(f o ) = {ρ Λ o b M(ρ)}, where imf = Λ(b). If P, Q R are regular partially ordered sets if f : P Q g : Q R are normal mappings, then (fg) o = f o g o. Let I Λ be regular partially ordered sets Γ : Λ I o : I Λ o are order preserving mappings, then (f, g) N(I) op N(Λ) is compatible with (Γ, ) if the following conditions hold: (c1) imf = I(x), img = Λ(y) kerf = Γ(y), kerg = (x) (c2) the diagram below commutes

4 174 P. G. Romeo P. R. Sreejamol I Λ o f I go Λ o Γ Λ I o g Λ Γ Theorem 1. ( Theorem 2 cf.[1] ) Let I, Λ be regular partially ordered sets Γ : Λ I o, : I Λ o be order preserving mappings. Then [I, Λ; Γ, ] is a cross connection if only if the following conditions are satisfied: (cr1) x M(Γ(y)) y M( (x)), x I, y Λ (cr2) If x M(Γ(y)) then the pair (ε I (Γ(y), x), ε Λ ( (x), y) is compatible with (Γ, ). Remark 1. U = U(I, Λ; Γ, ) consisting of all the pairs (f, g) that are compatible with (Γ, ) is a regular semigroup under the composition of mappings. Definition 6. An order preserving mapping f : P Q of posets is said to be residuated if there exists an order preserving mapping f + : Q P such that yf + f y x xff + for all x P y Q. The mapping f + is called the residual of f. The set ResP of all residuated maps of P is a semigroup f f + is a dual isomorphism. An f ResP is totally range closed if f maps principal ideals onto principal ideals. Observe that a residuated map that is also normal must be totally range closed, further f ResP is strongly range closed if f f + are totally range closed. The set B(P ) of all strongly range closed transformations of P is a sub semigroup of ResP. If f ResP if both f f + are normal, then f is binormal f B(P ). f o 3. Cross connection of Boolean lattice Recall that an ideal of a boolean lattice L is a subset I of L such that (1) 0 I, (2) if a I b I, then a b I, (3) if a I b L, then a b I. The principal ideal of L generated by a is L(a) = {b L b a} an ideal I of L is a complete ideal if {a i } is a family in I with a supremum a in L, then a I. Note that principal ideals are examples of complete ideals. Theorem 2. (Theorem 21 cf. [2]) The class of all complete ideals in a Boolean algebra L is itself a complete Boolean algebra with (1) 0 = {0}, (2) 1 = L, (3) M N = M N, (4) M N = {I : I is a complete ideal in L M N I}, (5) M c = {p L : p q = 0 for all q M}. I o

5 Cross connection of Boolean lattice 175 A subset of a Boolean lattice is a Boolean ideal if only if it is an ideal in the corresponding Boolean ring [2]. Proposition 2. Let L be a boolean lattice. For a L, let a c denotes the unique complement of a f a : L L f a + : L op L op defined by f a (x) = x a f a + (y) = y a c, x L, y L op. Then f a f + a are normal mappings. Proof. For x y, in L, clearly f a (x) f a (y) imf a = L(a), principal ideal generated by a. For every x L, there exists z = x a x such that L(z) = L(f a (x)) f a (z) = f a (x). Thus f a acts as an identity morphism from L(z) L(f a (x)). Thus f a is normal mapping. In a similar way it is seen that f a + is also normal. f a ResL is a binormal idempotent map which is residuated with residual f a + ResL op. Also f a f a + are totally range closed mappings such that f a B(L) f a + B(L op ). Further kerf a = {(x, y) x a = y a} kerf a + = {(x, y) x a = y a} Theorem 3. Let L be a boolean lattice a L. Define Γ(a) = kerf a (a) = kerf + a c so that Γ, are order preserving embedding from L (L op ) o L op L o respectively. Then Γ(a) = (a c ) for all a L [L, L op ; Γ, ] is a cross connection. Proof. We have imf a = L(a), imf a + = L op (a c ), kerf a + = Γ(a), kerf + = (a c ) Γ(a) = (a c ) hence kerf a = kerf a + the following diagrams commutes. Γ L (L op ) o f L Γ (f + ) o (L op ) o L op f + L op L o f o For (u, v) Γ(xf) u xf = v xf. f + (u xf) = f + (v xf), then f + u x = f + v x [follows from [6]]. Hence (f + u, f + v) Γ(x) thus (u, v) (f + ) 1 (Γ(x)) thus Γ(xf) (f + ) 1 Γ(x). Similarly, (f + ) 1 Γ(x) Γ(xf) so Γ(xf) = (f + ) o Γ(x) (f + y) = ( y)f o = ( y)f 1, thus (f, f + ) is compatible with (Γ, ). For a M(Γ(a)) there exists f a such that kerf a = Γ(a) a M(f a ) that is imf a = L(a), principal ideal generated by a. Dually, a c M( (a c )) means there exists f a + such that kerf a + = (a c ) a c M(f a + ). ie., imf a + = L op (a c ), principal ideal generated by a c. Thus a M(Γ(a)) a c M( (a c )), for every a L being binormal idempotent maps (f, f + ) is compatible with (Γ, ). Hence [L, L op ; Γ, ] is a cross connection. L o

6 176 P. G. Romeo P. R. Sreejamol Theorem 4. Let U = U(L, L op ; Γ, ) = {(f a, f + a ) f a B(L)} the set of all pairs of normal maps (f a, f + a ) with f a : L L, f + a : L op L op, that are compatible with (Γ, ). Clearly U B(L) B(L op ) U together with product defined by (f a, f + a ) (f b, f + b ) = (f a b, f + a b ) (f a, f a + ) + (f b, f b + ) = (f a b, f a b + ) where is the symmetric difference defined by a b = (a b c ) (b a c ). Then U(L, L op ; Γ, ) is a Boolean ring. Proof. (f a, f + a ) (f a, f + a ) = (f a, f + a ), ie., U is a b (f 0, f + 0 ) U is the additive identity [since L L op are boolean rings, with additive identity 0.] Every element in U is its own inverse, the addition in U is commutative for (f a, f + a ), (f b, f + b ), (f c, f + c ) U, (f a, f + a ) [(f b, f + b ) + (f c, f + c )] = (f a, f + a ) (f b c, f + b c ) = (f a (b c), f + a (b c) ) = (f (a b) (a c), f + (a b) a c) ) = (f a b, f + a b ) + (f a c, f + a c) = [(f a, f + a ) (f b, f + b )] + [(f a, f + a ) (f c, f + c ). [(f a, f + a ) + (f b, f + b )] (f c, f + c ) = [(f a, f + a ) (f c, f + c )] + [(f b, f + b ) (f c, f + c )] Hence U is a Boolean ring. Note that U is a Boolean lattice with respect to (f a, f + a ) (f b, f + b ) = (f a, f + a ) (f b, f + b ) (f a, f + a ) (f b, f + b ) = (f a, f + a ) + (f b, f + b ) + (f a, f + a ) (f b, f + b ) Lemma 1. (f a, f + a ) c = (f a c, f + a c) Proof. (f a, f a + ) (f a c, f a + c) = (f 0, f 0 + ) (f a, f a + ) (f a c, f a + c) = (f a a c, f a a + c) + (f 0, f 0 + ) = (f 1, f 1 + ) + (f 0, f 0 + ) = (f 1 0, f 1 0 ) = (f 1, f 1 ) hence (f a, f a + ) c = (f a c, f a + c)

7 Cross connection of Boolean lattice 177 Further, U is a strongly regular Baer ring the ideals I a = (f a, f + a ) of U are complete ideals generated by (f a, f + a ). The complete ideals of U form a complete Boolean lattice L the unique complement of I a is I a c, for every a, a c in L. Theorem 5. Let L be a boolean lattice U the cross connection ring obtained. Define a mapping ψ : L U as a (f a, f + a ), a L. Then ψ is a representation. Further it is one-one onto homomorphism. Hence L = U. Proof. ψ(a b) = (f a b, f + (a b) ) = (f a, f a + ) (f b, f b + ) = ψ(a) ψ(b) ψ(a b) = (f a b, f + (a b) ) = (f a f b, f a + op f b + ) = (f a, f a + ) (f b, f b + ) = ψ(a) ψ(b) ψ(a c ) = ((f a c, f a + c) = (f a, f a + ) c = ψ(a) c Thus ψ is a boolean isomorphism L = U. Similarly we get U = L. Hence L = U = L. Example 2. Let X = {a, b, c}, then L = (P (X), ) is a boolean lattice with elements φ, A = {a}, B = {b}, C = {c}, A c = {b, c}, B c = {a, c}, C c = {a, b} X. For S L, define f S : L L as x x S, x L f S + : L op L op as x x S c, x L op, where L op is the dual lattice of L. Then imf S = L(S) imf S + = L op (S c ), the principal ideals generated by S S c respectively. f S f S + are idempotent normal mappings f S is a residuated mapping with residual f S +. B(L) = {f S, S L} B(L op ) = {f S +, S L op }. X C c B c A c A B C φ Define Γ(S) = kerf S (S) = kerf S + the normal equivalence relations. Let c L o = {kerf S f S B(L)} (L op ) o = {kerf S + f S + B(L op )} then Γ(S) = (S c ) (f S, f S + ) is compatible with (Γ, ). Further it is easily seen that the conditions of cross connection are satisfied. Hence U = U([L, L op ; Γ, ]) = {(f S, f S + ) B(L) B(L op )} is the cross connection ring under the operations (f S, f S + ) (f T, f T + ) = (f S T, f S T + ) (f S, f S + ) + (f T, f T + ) = (f S T, f S T + ). U is

8 178 P. G. Romeo P. R. Sreejamol a strongly regular Baer ring isomorphic to the powerset ({a, b, c}, ). (f X, f + X) (f C c, f + C c) (f B c, f + B c) (f A c, f + A c) (f A, f + A ) (f B, f + B ) (f C, f + C ) (f φ, f + φ ) Again the complete ideals I S = (f S, f + S ) of U for every S L form a Boolean lattice L below is isomorphic to U. I X I C c I B c I A c I A I B I C I φ References [1] K.S.S. Nambooripad F.J. Pastijn, The fundamental representation of a strongly regular Baer semigroup, J. Algebra, 92 (1985), [2] Steven Givant Paul Halmos, Introduction to Boolean Algebras, Springer Verlag, New York, [3] P.A. Grillet, Semigroups, An Introduction to the Structure Theory, Tulane University, New Orleans, Louisiana, [4] P.A. Grillet, The structure of regular semigroups, I: A representation, Semigroup Forum, 8 (1974), [5] K.S.S. Nambooripad, Theory of cross connections, Publication No. 28, (1994), Centre for Mathematical Sciences, Trivrum.

9 Cross connection of Boolean lattice 179 [6] T.S. Blyth, Lattices Ordered Algebraic Structures, Springer Verlag, London, [7] K.M. Rangaswamy, Regular Baer rings, Proceedings of the American Mathematical Society, 42 (1974), no. 2, Received: April 23, 2017; Published: May 18, 2017