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The Lattice Structure of Ideals of a BCK-Algebra?
R.A. Borzooei and M. Bakhshi

The aim of this paper is to investigate some properties of the lattice of all ideals of a BCK-algebra and the interrelation among them; e.g, we show that BCK(X), the lattice of all ideals of a BCK-algebra, satisfies Join Infinitely Distributive identity, and under suitable conditions is Boolean. Moreover, we investigate the connection between atoms and ∨-irreducible elements in BCK(X). Then we prove that (BCK(X),∨,∧,* , 0, 1) is a semi-De Morgan algebra and also by considering the notion of pseudocomplement in a lattice, we prove that (P(BCK(X)),∨˙, ∧,* , 0, 1), where P(BCK(X)) is the set of all pseudocomplements in BCK(X), is a Boolean algebra.

2000 AMS Subject Classification: 06F35, 06B05

Keywords: BCK-algebra, algebraic lattice, relatively pseudocomplemented lattice, join infinite distributive lattice, Heyting algebra, Boolean algebra

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