Sets with two associative operation

TL;DR

This paper explores algebraic structures called dimonoids and duplexes, constructing free examples using combinatorial objects like planar trees and permutations, and establishing their freeness properties.

Contribution

It introduces a construction of free duplexes generated by sets, linking them to permutations and planar trees, and characterizes their freeness in specific varieties.

Findings

Permutations form a free duplex on a specific generator set.

Planar binary trees generate free duplexes with one generator.

Vertices of cubes also generate free duplexes in certain varieties.

Abstract

In this paper we consider dimonoids, which are sets equipped with two associative binary operations. Dimonoids in the sense of J.-L. Loday are xamples of duplexes. The set of all permutations, gives an example of a duplex which is not a dimonoid. We construct a free duplex generated by a given set via planar trees and then we prove that the set of all permutations form a free duplex on an explicitly described set of generators. We also consider duplexes coming from planar binary trees and vertices of the cubes. We prove that these duplexes are free with one generator in appropriate variety of duplexes.