Skew Boolean algebras and discriminator varieties

Cancellation in Skew Lattices

“…Cancellative skew lattices are of special interest since the two classes of skew lattices studied the most over the last twenty years, skew lattices in rings and skew Boolean algebras, are both cancellative. (See [1]- [4], [6], [9], [11], [12], [16] and [20].) Our main goals are (1) to show that all three classes of cancellative skew lattices are varieties and (2) to characterize each class by a short list of forbidden subalgebras, much as excluding M 3 and N 5 characterizes distributive lattices.…”

“…(See [1]- [4], [6], [9], [11], [12], [16] and [20].) Our main goals are (1) to show that all three classes of cancellative skew lattices are varieties and (2) to characterize each class by a short list of forbidden subalgebras, much as excluding M 3 and N 5 characterizes distributive lattices. This study is carried out in the five following sections, the next of which provides some background on skew lattices.…”

Rings Whose Idempotents Form a Multiplicative Set

“…Subsequent papers by Leech [12] and by Bignall and Leech [2] benefited from developments in skew lattices. Since then other papers have appeared on either skew Boolean algebras or their role in closely related topics.…”

“…That is, E(R) is a band (a semigroup of idempotents) on which xyzw = xzyw holds. For some time it has been known that any band S of idempotents in a ring that is maximal with respect to being normal is likewise closed under a counter-product, e∇f = (e • f ) 2 , that is also associative and idempotent. In this case S forms a noncommutative variant of a Boolean algebra called a skew Boolean algebra.…”

“…(See, e.g., [3], [5] and [14].) For further background on skew lattices and skew Boolean algebras see [2], [11] and [12].…”