I-semigroup
An $I$-semigroup [resp. $I$-monoid] is a semigroup $S$ [resp. a monoid $M$] with a unary operation $x\mapsto {x}^{-1}$ defined on $S$ [resp. on $M$] such that for each $x,y\in S$ [resp. for each $x,y\in M$]
$${({x}^{-1})}^{-1}=x,x=x{x}^{-1}x.$$ |
Notice that
$${x}^{-1}x{x}^{-1}={x}^{-1}{({x}^{-1})}^{-1}{x}^{-1}={x}^{-1},$$ |
so ${x}^{-1}$ is an inverse^{} of $x$.
The class of $I$-semigroups [resp. $I$-monoids] strictly contains the class of inverse semigroups [resp. inverse monoids]. In fact, the class of inverse semigroups [resp. inverse monoids] is precisely the class of $I$-semigroups with involution [resp. $I$-monoids with involution], i.e. the class of $I$-semigroups [resp. $I$-monoids] in which the unary operation ${}^{-1}$ is also an involution.
References
- 1 J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1991.
Title | I-semigroup |
---|---|
Canonical name | Isemigroup |
Date of creation | 2013-03-22 16:11:27 |
Last modified on | 2013-03-22 16:11:27 |
Owner | Mazzu (14365) |
Last modified by | Mazzu (14365) |
Numerical id | 5 |
Author | Mazzu (14365) |
Entry type | Definition |
Classification | msc 20M10 |
Related topic | SemigroupWithInvolution |
Defines | I-semigroup |
Defines | I-monoid |