Files

Download

Download Full Text (273 KB)

Description

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

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.

Copyright

Springer

On the Representation of Boolean Magmas and Boolean Semilattices

Included in

Algebra Commons

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.