Monoids

Part of speech: noun

Definitions

1. A specific type of algebraic construct featuring a set that includes an associative operation and an identity element to facilitate mathematical abstractions
2. A structure in mathematics defined by a set combined with an operation that is both associative and has a designated identity element, useful in theoretical contexts
3. A mathematical concept consisting of a set equipped with an operation that combines any two elements associatively and includes an element acting as a neutral identity

Etymology: "Monoids" is the plural of "monoid", so its history is that of the singular form. The base is the Greek prefix "mono-", meaning "single" or "alone", which English has used in countless compounds such as monorail and monologue. The suffix "-oid" comes from Greek "eidos", meaning "form" or "shape", and English uses it to mean "resembling" or "of the nature of", as in humanoid or asteroid. Put together, "monoid" literally suggests something like "single-form" or "resembling a single thing". The term is a mathematical coinage, not an ordinary word that evolved over centuries. In algebra, a monoid is a set equipped with an associative operation and an identity element, and the name reflects that such a structure has one operation, in contrast to the richer structures that mathematicians build on top of it.