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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.
ISBN
978-3-030-64187-0
Publication Date
6-1-2021
Publisher
Springer
Disciplines
Algebra
Recommended Citation
P. Jipsen, M. E. Kurd-Misto and J. Wimberley, On the representation of Boolean magmas and Boolean semilattices, In: Madarász J., Székely G. (eds) Hajnal Andréka and István Németi on Unity of Science. Outstanding Contributions to Logic, vol 19. Springer, Cham., (2021), Ch 12, 289-312. https://doi.org/10.1007/978-3-030-64187-0_12
Copyright
Springer
Comments
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.