Regular semigroup

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This article defines a semigroup property: a property that can be evaluated to true/false for any given semigroup
View a complete list of semigroup properties
QUICK PHRASES: semigroup with all elements regular, regular being a weak version of invertible


Symbol-free definition

A regular semigroup is a semigroup (i.e., a set with associative binary operation) in which every element is regular.

Definition with symbols

A semigroup (S,*) is termed a regular semigroup if every a \in S is regular, i.e.:

  • For every a \in S, there exists a b \in S such that aba = a
  • Equivalently, for every a \in S, there exists a c \in S such that aca = a and cac = c.

Relation with other properties

Stronger properties

External links