Counting finite linearly ordered involutive bisemilattices

TL;DR

This paper counts finite involutive bisemilattices, algebraic structures related to paraconsistent logic, by leveraging their representation as Plonka sums of Boolean algebras, specifically focusing on those with totally ordered semilattices.

Contribution

It provides a method to enumerate finite involutive bisemilattices up to isomorphism using their Plonka sum representation with totally ordered semilattices.

Findings

Count of finite involutive bisemilattices with totally ordered semilattices

Representation of these structures as Plonka sums of Boolean algebras

Enumeration results up to isomorphism

Abstract

The class of involutive bisemilattices plays the role of the algebraic counterpart of paraconsistent weak Kleene logic. Involutive bisemilattices can be represented as Plonka sums of Boolean algebras, that is semilattice direct systems of Boolean algebras. In this paper we exploit the Plonka sum representation with the aim of counting, up to isomorphism, finite involutive bisemilattices whose direct system is given by totally ordered semilattices.