semiframes

Semiframes are a concept in abstract algebra, specifically within the study of algebraic structures. A semiframe is a set equipped with two binary operations, typically denoted as addition (+) and multiplication (*), that satisfy certain distributive-like properties. These properties are weaker than those found in a ring, where multiplication is fully distributive over addition. In a semiframe, the multiplication operation might not be associative, and the distributive laws may only hold from one side, meaning either a * (b + c) = (a * b) + (a * c) for all elements a, b, and c, or (b + c) * a = (b * a) + (c * a) for all elements a, b, and c, or both. The exact axioms defining a semiframe can vary slightly depending on the context and the author, but they generally aim to capture structures that exhibit some, but not all, of the properties of a ring or a near-ring. An important distinction is that semiframes do not necessarily require an additive identity (zero element) or additive inverses. The study of semiframes is often motivated by generalizing existing algebraic structures and exploring their properties and relationships to other algebraic objects. Examples of structures that can be considered as semiframes include certain types of quasigroups or loops that also possess a multiplicative operation with some distributive behavior.