Semigroup collaborations between elementary operations

Abstract

Given two operations \(*\) and \(\circ \) on a set S, an operation \(\star \) on S is said to be a collaboration between \(*\) and \(\circ \) if for all \(a,b \in S\), \(a \star b\) \(\in \{a *b, a\circ b \}\). Another term for collaborations is two-option operations. We are interested in learning what associative collaborations of two given operations \(*\) and \(\circ \) there may be. We do not require that \(*\) and \(\circ \) themselves be associative. For this project, as an initial experiment, we consider Plus-Minus operations (i.e. collaborations between addition and subtraction on an abelian group) and Plus-Times operations (i.e. collaborations between the addition and multiplication operations on a semiring.) Our study of Plus-Minus operations focuses on the additive integers but extends to ordered groups. For Plus Times operations, we make some headway in the case of the semiring of natural numbers. We produce an exhaustive list of associative collaborations between the usual addition and multiplication on the natural numbers \({\mathbb {N}}\). The Plus-Times operations we found are all examples of a type of construction which we define here and that we call augmentations by multidentities. An augmentation by multidentities combines two separate magmas A and B to create another, A(B), having \(A \sqcup B\) as underlying set, and in such a way that the elements of B act as identities over those of A. Hence, B consists of a sort of multiple identities (explaining the moniker multidentities.) When A and B are both semigroups then so is A(B). Understanding the connection between certain collaborations and augmentation by multidenties removes, in several cases, the need for cumbersome computations to verify associativity. A final section discusses connections between group collaborations and skew braces.