The Localization of Commutative Bounded BCK-Algebras

In this paper we develop a theory of localization for bounded commutative BCK-algebras. We try to extend some results from the case of commutative Hilbert algebras (see [1]) to the case of commutative BCK-algebras.


Introduction
In 1966, Y. Imai and K. Iséki introduced a new notion called a BCK-algebra (see [2]).This notion is originated from two different ways.One of the motivations is based on the set theory (where the set difference operation play the main role) and another motivation is from classical and non-classical propositional calculi (see [2]).There are some systems which contain the only implication functor among the logical functors.These examples are the systems of positive implicational calculus, weak positive implicational calculus by A. Church, and BCI, BCK-systems by C. A. Meredith (see [3]).
In this paper we develop a theory of localization for commutative (bounded) BCK-algebras, and then we deal with generalizations of results which are obtained in the paper [1] for case of Hilbert algebras.For some informal explanations of the theory of localization for others categories of algebras see [4,5].
The paper is organized as follows: in Section 2 we recall the basic definitions and put in evidence many rules of calculus in (commutative) BCK-algebras which we need in the rest of paper.In Section 3 we introduce the commutative BCK-algebra of fractions relative to a ∨-closed system.In Section 4 we develop a theory for multipliers on a commutative (bounded) BCK-algebra.In Section 5 we define the notions of BCK-algebras of fractions and maximal BCK-algebra of quotients for a commutative (bounded) BCK-algebra.In the last part of this section is proved the existence of the maximal BCKalgebra of quotients (Theorem 29).In Section 6 we develop a theory of localization for commutative (bounded) BCK-algebras.So, for commutative (bounded) BCK-algebra A we define the notion of localization BCK-algebra relative to a topology F on A. In Section 7 we describe the localization BCK-algebra F A in some special instances.

Preliminaries
In this paper the symbols ⇒ and ⇔ are used for logical implication, respectively logical equivalence.

 
, ,1 A  of type (2,0) such that the following axioms are fulfilled for every , , x y z A  : (a 1 ) ; In [7] it is proved that the system of axioms {a 1 , a 2 , B, C, K} is equivalent with the system {a 2 , a 3 , a 4 , B}, where: (a 3 ) ; 1 1 x   (a 4 ) 1 x x   .For examples of BCK-algebras see [6][7][8].If A is a BCK-algebra, then the relation ≤ defined by x y  iff is a partial order on A (which will be called the natural order on A; with respect to this order 1 is the largest element of A. A will be called bounded if A has a smallest element 0; in this case for x   x for every , x y A  , then A is called commutative (see [5,9,10]).We have the following rules of calculus in a BCK-algebra A (see [6,7]): (c 1 ) x y x   ; , then for every , and Proposition 1 ([9], p. 5) If A is a commutative BCKalgebra, then relative to the natural ordering, A is a joinsemilattice, where for , x y A  : Lemma 2 Let A be a commutative BCK-algebra.For every , , x y z A  there exists     . Proof.Since x y x y   by (c 5 ) we deduce that . Let now such that .Then In [9] (Theorem 8) and [8] (Remark 2.1.32)it is proved the following result: Theorem 3 If A is a BCK-algebra, then the following assertions are equivalent: 1) For every , , x A BCK-algebra which verify one of the above equivalent conditions is called Hilbert algebra (or positive implicative BCK-algebra).
If A is a bounded BCK-algebra, we have the following rules of calculus in A (see [6]): that is, A is an involutive BCK-algebra (see [6], p. 115 and [9], p. 10).For 1 , , , , , ; . We denote by the set of all deductive systems of A. For a nonempty subset  We denote by   B A the set of all boolean elements of A; clearly, ([7]) Let A be a BCK-algebra.Then for every , x y A  , .
  Boolean elements also satisfy several interesting properties which can be proved using above corollary and some arithmetical calculus: is a Boolean algebra (where for , and .

 
Let A be a commutative BCK-algebra.For every and and from So to prove (*) let such that and Let A be a commutative BCK-algebra.Then for every and we have: f are bounded BCK-algebras, then we add the condition ). 0 0 

Commutative BCK-Algebra of Fractions Relative to a ∨-Closed System
In this section by A we denote a commutative bounded

Proposition 12  
A S is a bounded commutative BCKalgebra, when Remark 3 Since for every verify the following property of universality: Theorem 14 For every bounded commutative BCKalgebra B and every morphism of bounded BCK-algebras f A  B such that there exists a unique morphism of bounded BCK-algebras or is such that 0 S  and , then for is an ∨-closed system system such that (for example or ), then for every is called the BCK-algebra of fractions of A relative to S.

Multipliers on a Commutative Bounded BCK-Algebra
The concept of maximal lattice of quotients for a distributive lattice was defined by J. Schmid in [11,12] (taking as a guide-line the construction of complete ring of quotients by partial morphisms introduced by G. Findlay and J. Lambek (see [13], p. 36).The central role in the construction of the maximal lattice of quotients for a distributive lattice due to J. Schmidt in [11] and [12] is played by the concept of multiplier for a distributive lattice defined by W. H. Cornish in [14].
In this section we develop a theory for multipliers on a commutative bounded BCK-algebra A.
the set of all ∨-subsets of A.

Clearly
(and more generally, if denote by the set of all increasing subsets of A, then the languag we wil m To simplify ins ps 0 e, l use strong multiplier ,1 tead partial strong multiplier using total to indicate that the domain of a certain multiplier is A. Examples The ma and respectively 1   1 x  , for every x A  are total strong multipliers on A. 2) For  Indeed, if in (sm 1 ) we put e for every e x e y     .
For every Proof.The fact th is a commutative BCKalgebra follows from Lem .If

 ,
x do that is, , e e x e e y e e x e e y e x e y x y and suppose by contrary that there exists  , ). n upper for the family : mediately; to rove the transitivity o p , or every for every hence for 0 x  we obtain that 0

 .
D T A  Thus, D D and for every , f is principal which is contradictory with the assum tion that p f is non-principal.□

mmutative BCK-Algebra of Quotients
Th section is to define (taking as a guidene the case of distributive lattices) the notions of BCK-

Maximal Co
e goal of this li algebra of fractions and maximal BCK-algebra of quotients for a commutative bounded BCK-algebra.For some informal explanations of notions of fraction see [13] and [5].
Definition 10 A bounded commutative BCK-algebra A is called BCK-algebra of fractions of A if: e x e e x x e x x e x x If A is a Boolean algebra, then   .

B A A 
By Proposition 26,   Q A is a Boolean alge ax bra and the ioms sm 1 -sm 4 are equivalent with sm 1 , hence   Q A is in this case just the cal Dedekind-MacNeille com-pletion of A (see [12], p. 687).In contrast to th ral situation, the Dedekind-MacNeille completion of a Booalgebra is again distributive and, in fact, is a Boolean algebra (see [15] Proof.
by Lemm ce a 27 (sin recall tha is a topological system on A. denote the set of all regular subsets of Example 4 We t by   R A we A (see Definition 8).Then F is a topological system on A. is a topological system on A, l F If et us consider the relation F  of A defined by: there exists x A  and by : of a distributive lattice L with respect to a topology F on see L in a similar way as for rings (see [16]) or monoids ( [17]).The aim of this section is to define the notion of localization BCK-algebra F A of a commutative bounded BCK-algebra A with respect to a topology F on A. In the last part of this section proved that the maximal commutative BCK-algebra of quotients (defi d in Section 5) and the commutative BCK-algebra of fractions relative to a ∨-closed system (defined in Section 3) are BCK-algebras of localization.
In this section A will be a bounded commutative BCK-algebra and is ne F a topological system on A. Definition 12 A non-empty family F of elements on   T A will be called a topological system on A if the following properties hold:  and denote by the in ctive limit (in the category du of sets): For any F-multiplier : This definition is correct.Indeed, let : ). Definition 14 F A will be called the localization BCKalgebra of A with respect to the topology F .
Lemma 32 The mapping  

 
To obtain the maximal BCK-algebra of quotients   Q A as a loc ization relative to l system al a topologica F we will develop another theory of F-multipliers (m Def str eaning we add new axioms for F-multipliers).
inition 15 Let F be a topological system on A. A ong-F-multiplier is a mapping :

Re
If A is a BCK-algebra, the maps 0,1:   .
We shall prove that

F
verifies the axioms m 1 and m 2 and ( Let A be a BCK-algebra.A deductive system (or i-filter) of A is a nonempty subset D of A such that1 D for some , , * .** y  * *, .
More general, if A is a BCK-algebra such that  is injective.To prove the surjectivity of  , let