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A magma is an algebra with a binary operation ·, and a Boolean magma is a Boolean algebra with an additional binary operation · that distributes over all finite Boolean joins. We prove that all square-increasing (x x2) Boolean magmas are embedded in complex algebras of idempotent (x = x2) 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 · is associative, commutative, and square-increasing. Let SL be the class of semilattices and let S(SL+) be all subalgebras of complex algebras of semilattices. All members of S(SL+) are Boolean semilattices and we investigate the question of which Boolean semilattices are representable, i.e., members of S(SL+). There are 79 eight-element integral Boolean semilattices that satisfy a list of currently known axioms of S(SL+). We show that 72 of them are indeed members of S(SL+), leaving the remaining 7 as open problems.



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This is a pre-copy-editing, author-produced PDF of a chapter accepted for publication in Judit Madarász and Gergely Székely (Eds.), Hajnal Andréka and István Németi on Unity of Science: From Computing to Relativity Theory Through Algebraic Logic. This version may not exactly replicate the final publication.



On the Representation of Boolean Magmas and Boolean Semilattices

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Algebra Commons



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