Boolean representation of bounded BCK-algebras

Abstract

We define the Boolean center and the Boolean skeleton of a bounded BCK-algebra, and we use the Boolean skeleton to obtain a representation of bounded BCKalgebras, called (weak) Pierce bBCK-representation, as (weak) Boolean products of bounded BCK-algebras. We analyze the cases in which the stalks in these representations are directly indecomposable, finitely subdirectly irreducible or simple algebras. We give some examples of algebras and relative subvarieties of bounded BCK-algebras to illustrate the results.

Semisimplicity and the discriminator in bounded BCK-algebras

“…Let A ∈ V be subdirectly irreducible. Then, by Corollary 1.5 of [10], (EM n ) implies that for any a ∈ A {1}, a n → 0 = 1, and so A is simple. Therefore V is semisimple.…”

“…There is a great deal of literature of BCK-algebras, but the references given by W. Blok and J. G. Raftery in [2,3] are sufficiently representative. For arithmetical properties on bounded BCK-algebras we also use results given in [6,9,10].…”

The Localization of Commutative Bounded BCK-Algebras

“…x y x    . 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:…”

“…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  , then A is called commutative (see [5,9,10]). We have the following rules of calculus in a BCK-algebra A (see [6,7]   ** ** ** , . …”