On multisemigroups

TL;DR

This paper explores the fundamental properties of multisemigroups, providing new examples, constructions, and insights into their structure, especially focusing on simple and nilpotent cases, and demonstrating their prevalence in random operations.

Contribution

It offers a comprehensive study of multisemigroups, including new examples, constructions, and analysis of their properties, with emphasis on simple and nilpotent types, and probabilistic existence results.

Findings

Many new and old examples of multisemigroups

Almost all random multivalued operations form multisemigroups

Detailed analysis of simple and nilpotent multisemigroups

Abstract

Motivated by recent appearance of multivalued structures in categorification, tropical geometry and other areas, we study basic properties of abstract multisemigroups. We give many new and old examples and general constructions for multisemigroups. Special attention is paid to simple and nilpotent multisemigroups. We also show that "almost all" randomly chosen multivalued binary operations define multisemigroups.