Semilattice Ordered Inverse Semigroups

Abstract

In this expository talk we give some recent results on the structure of inverse semigroups endowed with a compatible semilattice ordering. In addition we consider some open questions regarding these semigroups.

Semiring identities of the Brandt monoid

“…Recall that an additively idempotent semiring an algebra (S, +, •) of type (2, 2) such that the additive reduct (S, +) is a semilattice (that is, a commutative idempotent semigroup), the multiplicative reduct (S, •) is a semigroup, and multiplication distributes over addition on the left and on the right, that is, (S, +, •) satisfies the identities x(y + z) ≈ xy + xz and (y + z)x ≈ yx + zx. In papers which motivation comes from semigroup theory, objects of this sort sometimes appear under the name semilattice-ordered semigroups, see, e.g., [8] or [12]. We will stay with the term 'additively idempotent semiring', abbreviated to 'ai-semiring' in the sequel.…”