Sets with two associative operation
Teimuraz Pirashvili

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Algebra and Logic · Advanced Combinatorial Mathematics
