# On noncommutative generalisations of Boolean algebras

## TL;DR

This paper explores noncommutative generalizations of Boolean algebras, introducing skew Boolean algebras and Boolean-like algebras, establishing their structural relationships, and developing a theory of multideals and representations.

## Contribution

It introduces skew star algebras as a new variety term equivalent to nBAs and develops a general theory of multideals and representations for these structures.

## Key findings

- Any nBA is a cluster of n isomorphic right-handed skew BAs.
- The variety of skew star algebras is term equivalent to nBAs.
- Representation theorem for right-handed skew BAs in terms of n-partitions.

## Abstract

Skew Boolean algebras (skew BA) and Boolean-like algebras (nBA) are one-pointed and n-pointed noncommutative generalisation of Boolean algebras, respectively. We show that any nBA is a cluster of n isomorphic right-handed skew BAs, axiomatised here as the variety of skew star algebras. The variety of skew star algebras is shown to be term equivalent to the variety of nBAs. We use skew BAs in order to develop a general theory of multideals for nBAs. We also provide a representation theorem for right-handed skew BAs in terms of nBAs of n-partitions.