An

(The convention followed in this article will be that of writing a function on the right of its argument, andcomposing functions from left to right - a convention often observed in semigroup theory.)

- dom αβ = α $cap$ dom βα
^{-1}

There are a number of equivalent characterisations of an inverse semigroup

- Every element of
*S*has a unique inverse, in the above sense. - Every element of
*S*has at least one inverse (*S*is a regular semigroup) and idempotents commute (that is, the idempotents of*S*form a semilattice). - Every $mathcal\{L\}$-class and every $mathcal\{R\}$-class contains precisely one idempotent, where $mathcal\{L\}$ and $mathcal\{R\}$ are two of Green's relations.

- $a,mathcal\{L\},bLongleftrightarrow\; a^\{-1\}a=b^\{-1\}b,quad\; a,mathcal\{R\},bLongleftrightarrow$

Examples of inverse semigroups:

- Every group is an inverse semigroup.
- The bicyclic semigroup is inverse, with (
*a*,*b*)^{-1}= (*b*,*a*). - Every semilattice is inverse.
- The Brandt semigroup is inverse.
- The Munn semigroup is inverse.

Unless stated otherwise,

- $a\; leq\; b\; Longleftrightarrow\; a=eb,$

- $a\; leq\; b\; Longleftrightarrow\; a=bf,$

The natural partial order is compatible with both multiplication and inversion, that is,Howie 1995 :152-3

- $a\; leq\; b,\; c\; leq\; d\; Longrightarrow\; ac\; leq\; bd$

- $a\; leq\; b\; Longrightarrow\; a^\{-1\}\; leq\; b^\{-1\}.$

In a group, this partial order simply reduces to equality, since the identity is theonly idempotent. In a symmetric inverse semigroup, the partial order reduces to restriction of mappings,i.e., α ≤ β if, and only if, the domain of α is contained in the domain of β and

The natural partial order on an inverse semigroup interacts with Green's relations as follows: if

On

- $e\; leq\; f\; Longleftrightarrow\; e\; =\; ef,$

One of the earliest results proved about inverse semigroups was the

- dom φ =
*Sa*^{-1}and*x*(*a*φ) =*xa*

Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup.

- $a,rho,b,quad\; c,rho,dLongrightarrow\; ac,rho,bd.$Howie 1995 22

Of particular interest is the relation $sigma$, defined on an inverse semigroup

- $a,sigma,bLongleftrightarrow$ there exists a $cin\; S$ with $cleq$

A congruence ρ on an inverse semigroup

- $ain\; S,\; ein\; E(S),\; a,rho,eLongrightarrow\; ain\; E(S).$Lawson 1998 65

- $es\; in\; E\; Longrightarrow\; s\; in\; E.$

- $se\; in\; E\; Rightarrow\; s\; in\; E.$Howie 1995 192.

One further characterisation of an

*S*is*E*-unitary;- σ is idempotent pure;
- $sim$ = σ,

- $asim\; bLongleftrightarrow\; ab^\{-1\},a^\{-1\}b$ are idempotent.

Central to the study of

- $mathcal\{Y\}$ is a lower semilattice, that is, every pair of elements
*A*,*B*in $mathcal\{Y\}$ has a greatest lower bound*A*$wedge$*B*in $mathcal\{Y\}$ (with respect to ≤); - $mathcal\{Y\}$ is an order ideal of $mathcal\{X\}$, that is, for
*A*,*B*in $mathcal\{X\}$, if*A*is in $mathcal\{Y\}$ and*B*≤*A*, then*B*is in $mathcal\{Y\}$.

Now let

- for all
*g*in*G*and all*A*,*B*in $mathcal\{X\}$,*gA*=*gB*if, and only if,*A*=*B*; - for each
*g*in*G*and each*B*in $mathcal\{X\}$, there exists an*A*in $mathcal\{X\}$ such that*gA*=*B*; - for all
*A*,*B*in $mathcal\{X\}$,*A*≤*B*if, and only if,*gA*≤*gB*; - for all
*g*,*h*in*G*and all*A*in $mathcal\{X\}$,*g*(*hA*) = (*gh*)*A*.

The triple $(G,\; mathcal\{X\},\; mathcal\{Y\})$ is also assumed to have the following properties:

- for every
*X*in $mathcal\{X\}$, there exists a*g*in*G*and an*A*in $mathcal\{Y\}$ such that*gA*=*X*; - for all
*g*in*G*,*g*$mathcal\{Y\}$ and $mathcal\{Y\}$ have nonempty intersection.

Such a triple $(G,\; mathcal\{X\},\; mathcal\{Y\})$ is called a

- $P(G,\; mathcal\{X\},\; mathcal\{Y\})\; =\; \{\; (A,g)\; in\; mathcal\{Y\}times\; G\; g^\{-1\}A\; in\; mathcal\{Y\}\; \}$

- $(A,g)(B,h)=(A\; wedge\; gB,\; gh)$.

Examples of regular generalisations of an inverse semigroup are:Howie 1995 : Section 2.4 & Chapter 6.

*Regular semigroups*: a semigroup*S*is*regular*if every element has at least one inverse; equivalently, for each*a*in*S*, there is an*x*in*S*such that*axa*=*a*.*Locally inverse semigroups*: a regular semigroup*S*is*locally inverse*if*eSe*is an inverse semigroup, for each idempotent*e*.*Orthodox semigroups*: a regular semigroup*S*is*orthodox*if its subset of idempotents forms a subsemigroup.*Generalised inverse semigroups*: a regular semigroup*S*is called a*generalised inverse semigroup*if its idempotents form a normal band, i.e.,*xyzx*=*xzyx*, for all idempotents*x*,*y*,*z*.

The class of generalised inverse semigroups is the intersection of the class oflocally inverse semigroups and the class of orthodox semigroups.Howie 1995 : 222.

Amongst the non-regular generalisations of an inverse semigroup are:Fountain1979.[1]

- (Left, right, two-sided) adequate semigroups.
- (Left, right, two-sided) ample semigroups.
- (Left, right, two-sided) semiadequate semigroups.
- Weakly (left, right, two-sided) ample semigroups.

- A. H. Clifford and G. B. Preston,
*The Algebraic Theory of Semigroups*, Volume 2, Mathematical Surveys of the American Mathematical Society, No. 7, Providence, R.I., 1967. - V. Gould, "(Weakly) left E-ample semigroups"
- J. M. Howie,
*Fundamentals of Semigroup Theory*, Clarendon Press, Oxford, 1995. - M. V. Lawson,
*Inverse Semigroups: The Theory of Partial Symmetries*, World Scientific, 1998. - M. Petrich,
*Inverse semigroups*, Wiley, New York, 1984.

- Pseudogroup
- Partial symmetries
- Regular semigroup
- Semilattice
- Green's relations
- Category theory
- Inductive groupoid