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.
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Jipsen, P., Eyad Kurd-Misto, M., Wimberley, J. (2021). 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. https://doi.org/10.1007/978-3-030-64187-0_12
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