Pseudo-BCK Algebras: An Extension of BCK Algebras

Abstract

Abstract. We extend BCK algebras to pseudo-BCK algebras, as MV algebras and BL algebras were extended to pseudo-MY algebras and pseudo-BL algebras, respectively. We make the connection with pseudo-MV algebras and with pseudo-BL algebras.

On a theorem of Cantor-Bernstein type for algebras

“…[1]) and, more generally, the class of all pseudo M V -algebras (cf. Georgescu and Iorgulescu [6], [7], and Rachůnek [22]; in [22] the term 'generalized M V -algebra' was applied) are examples of L-varieties.…”

Pseudo BCK-Semilattices

“…It was proved in [6] that bounded commutative pseudo BCK-algebras (called here lattice-ordered pseudo BCK-algebras) are termwise equivalent to pseudo MV-algebras-non-commutative generalizations of MV-algebras introduced by G. Georgescu and A. Iorgulescu [5] and independently by J. Rachunek [16]. The equivalence with the standard signature {©," , 0,1} is given as follows: if (A, -0,1) is a bounded commutative pseudo BCKalgebra and we put x © y = (x 0) -• y = (y -• 0) x, x~ = x -> 0 and = x 0, then (A, ffi,~ , 0,1) is a pseudo MV-algebra, and the reverse passage from (A, ©,~ , 0,1) to (A, -0,1) is given by x -> y = x~ © y and x y = y © .…”

“…In the logical context this means that the strong conjunction is not commutative and the implication splits into two ones. Accordingly, G. Georgescu and A. Iorgulescu [6] introduced pseudo BCK-algebras as an extension of BCK-algebras: DEFINITION 1.1. A structure {A, <, -1), where < is a binary relation on A, -> and are binary operations on A, and 1 is a distinguished element of A, is called a pseudo BCK-algebra (pedantically, a reversed left pseudo BCK-algebra [10]) if it satisfies the following axioms, for all x,y,z € A:…”

Bounded pseudo-hoops with internal states

“…We mention that the above notations differ from the ones introduced in [38], but we use them to be in line with other works ( [37], [9], [8]). Note that in [8], x ∨ 1 y and x ∨ 2 y defined in [38] were replaced for the same reason with the notations:…”