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Morphisms on Closure Spaces and Moore Spaces B International Journal of Pure and Applied Mathematics Volume 91 No. 2 2014, 197-207 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v91i2.6 ijpam.eu MORPHISMS ON CLOSURE SPACES AND MOORE SPACES B. Venkateswarlu1 §, R. Vasu Babu2, Getnet Alemu3 1Department of Mathematics GIT, GITAM University Visakhapatnam, 530 045, A.P., INDIA 2Department of Mathematics Shri Vishnu Engineering college for Women Bhimavaram, 534201, W.G. Dist., A.P., INDIA 3Department of Mathematics Gondar University Gondar, ETHIOPIA Abstract: We discuss an equivalence of the concepts of complete lattices, Moore classes, and a one-to-one correspondence between Moore classes of sub- sets of a set X and closure operators on X. Also, we establish a correspon- dence between closure operators on sets and complete lattices. We describe morphisms among partially ordered sets, lattices, Moore classes, closure opera- tors and complete lattices and discuss certain inter-relationships between these objects. AMS Subject Classification: 06A15, 06F30, 54H12 Key Words: closure operator, Moore class, compact element, complete lat- tice, algebraic Lattice, frame, morphisms 1. Introduction and Preliminaries A partially ordered set (poset) is a pair (X, ≤), where X is a non empty set and c 2014 Academic Publications, Ltd. Received: September 26, 2013 url: www.acadpubl.eu §Correspondence author 198 B. Venkateswarlu, R.V. Babu, G. Alemu ≤ is a partial order (a reflexive, transitive and antisymmetric binary relation) on X. For any subset A of X and x ∈ X, x is called a lower bound (upper bound) of A if x ≤ a (a ≤ x respectively) for all a ∈ A. A poset (X, ≤) is called a lattice if every nonempty finite subset of X has greatest lower bound ( or glb or infimum ) and least upper bound ( or lub or supremum) in X. If (X, ≤) is a lattice and, for any a, b ∈ X, if we define a ∧ b = infimum {a, b} and a ∨ b = supremum {a, b}, then ∧ and ∨ are binary operations on X which are commutative, associative and idempotent and satisfy the absorption laws a∧(a∨b) = a = a∨(a∧b). Conversely, any algebraic system (X, ∧, ∨) satisfying the above properties becomes a lattice in which the partial order is defined by a ≤ b ⇐⇒ a = a ∧ b ⇐⇒ a ∨ b = b. A lattice (X, ∧, ∨) is called distributive if a∧(b∨c) = (a∧b)∨(a∧c) for all a, b, c ∈ X ( equivalently a∨(b∧c) = (a∨b)∧(a∨c) for all a, b, c ∈ X). A lattice (X, ∧, ∨) is called a bounded lattice if it has the smallest element 0 and largest element 1; that is, there are elements 0 and 1 in X, such that 0 ≤ x ≤ 1 for all x ∈ X. A partially ordered set in which every subset has infimum and supremum is called a complete lattice. If (L, ≤) is a complete lattice and X ⊆ L, we write infX or ∧X or V x for the infimum of X and sup X or ∨X or W x for the x∈X x∈X n supremum of X. If X = {x1, x2, ··· , xn} is a finite subset, then we write V xi i=1 n or x1 ∧ x2 ∧ · · · ∧ xn for infX and W xi or x1 ∨ x2 ∨ · · · ∨ xn for supX. Any i=1 complete lattice has the smallest element and the greatest element which are denoted by 0 and 1 respectively.Logically, the infimum and supremum of the empty set are 1 and 0 respectively. An element a 6= 0 in a complete lattice L is called compact if, for any A ⊆ L, a ≤ supA =⇒ a ≤ supF for some finite F ⊆ A. A complete lattice in which every element is the supremum of a set of compact elements is called an algebraic lattice. For elementary properties of posets and lattices we refer to [ 1 ] and [ 2 ]. 2. Moore Classes and Closure Operators In this section, we introduce the notion of a Moore class and discuss certain important elementary properties of these. To begin with, we have the following. Definition 2.1. Let X be any non-empty set. A non-empty class M of subsets of X is called a Moore class on X if M is closed under arbitrary M intersections, in the sense that, if {Mα}α∈∆ is a subclass of , then T Mα ∈ α∈∆ MORPHISMS ON CLOSURE SPACES AND MOORE SPACES 199 M . Example 2.2. Let X be a topological space and C be the class of all closed subsets of X [ 3 ]. Then C is a Moore class on the set X. Example 2.3. Let G be a group and S be the class of all subgroups of G. Then S is a Moore class on the set G. Example 2.4. Let A be any universal algebra [ 4 ],where there is atleast one fundamental nullary operation, and let S be the class of all subalgebras of A. If a0 is an element of A corresponding to a fundamental nullary operation on A, then a0 belongs to every subalgebra of A. From this it follows that the intersection of any class of subalgebras of A is non-empty and hence a subalgebra of A. Thus S is a Moore class on A. Example 2.5. For any non-empty set X, the whole power set P(X) is a Moore class on X and is called the discrete Moore class on X. Example 2.6. Let X be a non-empty set and A ⊆ X. Then the class {A, X} is a Moore class on X. Since the intersection of the empty class of subsets of a set X is the whole set X, it follows that any Moore class necessarily contains X. The following is a straight forward verification. Theorem 2.7. Let M be a Moore class on a set X. For any subset A of X, define A = ∩{M ∈ M | A ⊆ M}. Then the following hold for any subsets A and B of X (1). A ⊆ A (2). A = A (3). A ⊆ B =⇒ A ⊆ B (4). A is the smallest member in M containing A. Corollary 2.8. If M is a Moore class on X, then M = {A ⊆ X | A = A}. Note that in 2.2 above, A is the topological closure of A, while in 2.3, A is the subgroups generated by A. Definition 2.9. Let X be a non-empty set and P(X) the set of all subsets of X. A mapping c : P(X) → P(X) is called a closure operator on X if it satisfies the following for any A and B in P(X) 200 B. Venkateswarlu, R.V. Babu, G. Alemu (1). c is extensive ; that is, A ⊆ c(A) (2). c is idempotent ; that is, c(c(A)) = c(A) (3). c is inclusion preserving ; that is, A ⊆ B ⇒ c(A) ⊆ c(B). The following is an immediate consequence of Theorem 2.7. Theorem 2.10. Let M be a Moore class on a set X and, for any A ⊆ X, define cM (A) = ∩ {M ∈ M | A ⊆ M}. Then cM is a closure operator on X. The following is routine verification. Theorem 2.11. Let c be a closure operator on a set X and Mc = {A ⊆ X | c(A) = A} = {c(A) | A ⊆ X}. Then Mc is a Moore class on X. Also, c 7→ Mc is a one-to-one correspondence between the closure operators on X and Moore classes on X. Definition 2.12. A closure operator c on a set X is called a topological closure operator if c(φ) = φ and c(A ∪ B) = c(A) ∪ c(B) for any subsets A and B of X. Theorem 2.13. Let c be a closure operator on a set X and Mc be the Moore class on X corresponding to c, as given in Theorem 2.11. Then c is a topological closure operator on X if and only if Mc is closed under finite unions, M M in the sense that, for any finite subclass {Mi}i∈I of c, S Mi ∈ c. i∈I Proof. Suppose that c is a topological closure operator on X. First, let us observe that the empty set φ belongs to Mc, since c(φ) = φ. Let {Mi}i∈I be a finite subclass of Mc. If I is empty, then [ Mi = φ ∈ Mc i∈I Therefore, we can assume that I is non-empty, say I = {1, 2, . , n}, where n is a positive integer. Then [ Mi = M1 ∪ M2 ∪ ... ∪ Mn i∈I = c(M1) ∪ c(M2) ∪ ... ∪ c(Mn) , sinceMi ∈ Mc MORPHISMS ON CLOSURE SPACES AND MOORE SPACES 201 = c(M1 ∪ M2 ∪ ... ∪ Mn) M M and therefore S Mi ∈ c. Thus c is closed under finite unions. i∈I Conversely, suppose that Mc is closed under finite unions. In particular, φ ∈ Mc, since the union of the empty class of sets is empty. Therefore c(φ) = φ. Also, for any subsets A and B of X, we have c(c(A)) = c(A) and c(c(B)) = c(B) and hence c(A) and c(B) are members of Mc. Since Mc is closed under finite unions, it follow that c(A) ∪ c(B) ∈ Mc and therefore c(c(A) ∪ c(B)) = c(A) ∪ c(B). Now, c(A) ∪ c(B) ⊆ c(A ∪ B) ⊆ c(c(A) ∪ c(B)) = c(A) ∪ c(B) and hence c(A∪B) = c(A)∪c(B). Thus c is a topological closure operator. The following can be easily verified. Theorem 2.14. Let c be a closure operator on a set X.
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