Convex Congruences on BCK-Algebras

Tadeusz Traczyk, Wiesław Zarębski
1985 Demonstratio Mathematica  
Dedicated to the memory of Professor Roman Sikorski Introdaction In this note we shall prove that all thois-3 (and only those) congruence relations on BCK-algebras are convex which keep their factor algebras being again BCK-algebras, and that any convex congruence class V of BCK-algebras has congruence extension property if and only if S(V) is again a convex congruenoe class 5 in particular each variety of BCK-algebras does have congruenoe extension property. An algebra A = (A; •, 0J of type
more » ... 0) is said to be a BCK-algebra (see e.g. K. Iseki [4]) provided, for all x,y,z in A (we put dot only to avoid first-order brackets and use juxtaposition in other cases): BCK1. (xyxz).zy = 0, BCK2. (x»xy)y = 0, BCK3. xx = 0, BCK4. Ox = 0, BCK5. xy = 0 = yx implies x = y. The underlying set A is partially ordered by the relation x«,y if and only if xy = 0. If the set jxeA: xa=£b} has a largest element, for some a,b e A, it is denoted by a+b and called a sum. If the sum a+b axists for all in A, we say that the BCK-algebra A. has a sua (sea 9,. K. Iseki, ! 5j).
doi:10.1515/dema-1985-0123 fatcat:6wfo4rubgrgbzg4epyewaxkh4i