On the Representation of Boolean Magmas and Boolean Semilattices

Abstract

A magma is an algebra with a binary operation $\cdot$, and a Boolean magma is a Boolean algebra with an additional binary operation $\cdot$ that distributes over all finite Boolean joins. We prove that all square-increasing ($x\le x^2$) Boolean magmas are embedded in complex algebras of idempotent ($x=x^2$) magmas. This solves a problem in a recent paper [3] by C. Bergman. Similar results are shown to hold for commutative Boolean magmas with an identity element and a unary inverse operation, or with any combination of these properties. A Boolean semilattice is a Boolean magma where $\cdot$ is associative, commutative, and square-increasing. Let $\mathsf {SL}$ be the class of semilattices and let $\boldsymbol{\mathsf S}(\mathsf {SL}^+)$ be all subalgebras of complex algebras of semilattices. All members of $\boldsymbol{\mathsf S}(\mathsf {SL}^+)$ are Boolean semilattices and we investigate the question which Boolean semilattices are representable, i.e., members of $\boldsymbol{\mathsf S}(\mathsf {SL}^+)$. There are 79 eight-element integral Boolean semilattices that satisfy a list of currently known axioms of $\boldsymbol{\mathsf S}(\mathsf {SL}^+)$. We show that 72 of them are indeed members of $\boldsymbol{\mathsf S}(\mathsf {SL}^+)$, leaving the remaining 7 as open problems.