Monoid property

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This article is about a general term. A list of important particular cases (instances) is available at Category:Monoid properties

Definition

Symbol-free definition

A monoid property is a map from the collection of all monoids to the two-element set (true, false) with the property that any two isomorphic monoids get mapped to the same thing.

Caution

A monoid property must be decidable purely based on the abstract monoid structure, and should not be dependent on additional structure. Even if such additional structure occurs in the definition, it should have a universal or existential quantification associated to it.

Examples

Important examples of monoid properties

Being commutative is a monoid property: a monoid is commutative if any two elements in it commute. Every monoid either is commutative or is not commutative.