The Hadamard multiary quasigroup product

TL;DR

This paper explores the properties and generalizations of the Hadamard quasigroup product, extending it to multiary groupoids and quasigroups, and analyzing its algebraic preservation and combinatorial implications.

Contribution

It introduces a generalized version of the Hadamard quasigroup product for multiary structures and investigates its algebraic and combinatorial properties.

Findings

The operator preserves algebraic identities and structures.

The number of m-ary quasigroups equals the number of orthogonal m-ary operations.

Generalization to multiary quasigroups broadens the applicability of the product.

Abstract

The Hadamard quasigroup product has recently been introduced as a natural generalization of the classical Hadamard product of matrices. It is defined as the superposition operator of three binary operations, one of them being a quasigroup operation. This paper delves into the fundamentals of this superposition operator by considering its more general version over multiary groupoids. Particularly, we show how this operator preserves algebraic identities, multiary groupoid structures, inverse elements, isotopes, conjugates and orthogonality. Then, we generalize the mentioned Hadamard quasigroup product to multiary quasigroups. Based on this product, we prove that the number of $m$-ary quasigroups defined on a given set $X$ coincides with the number of $m$-ary operations that are orthogonal to a given $m$-set of orthogonal $m$-ary operations over $X$.