On skew Heyting algebras

Abstract

In the present paper we generalize the notion of a Heyting algebra to the non-commutative setting and hence introduce what we believe to be the proper notion of the implication in skew lattices. We list several examples of skew Heyting algebras, including Heyting algebras, dual skew Boolean algebras, conormal skew chains and algebras of partial maps with poset domains.

Skew Semi-Heyting Algebras

“…Clearly (e) implies that x → y ∈ y↑⊆ b ↑. Thus the restriction b → of → to b↑ is well defined(see [6]). Since b↑ is commutative (a), (b) and, (c) and (d) for → simplify respectively to (SH4), (SH2) and (SH3) for b → making b → is the binary operation on b↑.…”

“…If a skew lattice is strongly distributive, then it is normal. Dually if a skew lattice is costrongly distributive, then it is conormal (see [3], [6]). …”

“…Lemma 2.9 ( [6]): Let (L, ∨, ∧, →, 1) be a skew Heyting algebra and let x, y, a ∈ L be such that x, y ∈a↑ hold. Then x → y = x → a y.…”

“…Definition 2.8 ( [6]): An algebra L = (L; ∨, ∧, →, 1) of type (2, 2, 2, 0) is said to be a skew Heyting algebra whenever the following conditions are satisfied:…”

“…While Boolean algebras provide algebraic models of classical logic, Heyting algebras provide algebraic models of intuitionistic logic. The notion of skew Heyting algebra was introduced by Karin Cvetko-vah [6] as a generalization of Heyting algebra. In that paper, it is proved that a skew Heyting algebras form a variety and that the maximal lattice image of a skew Heyting algebra is a generalized Heyting algebra.…”