### Inverse semigroups : the theory of partial symmetries

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McAlister's P -theorem has been used to characterize F -inverse semigroups as well. A McAlister triple is an F -inverse semigroups if and only if is a principal ideal of and is a semilattice. A construction similar to a free group is possible for inverse semigroups. A presentation of the free inverse semigroup on a set X may be obtained by considering the free semigroup with involution , where involution is the taking of the inverse, and then taking the quotient by the Vagner congruence. The word problem for free inverse semigroups is much more intricate than that of free groups. A celebrated result in this area due to W.

Munn who showed that elements of the free inverse semigroup can be naturally regarded as trees, known as Munn trees.

### Module Details

Multiplication in the free inverse semigroup has a correspondent on Munn trees, which essentially consists of overlapping common portions of the trees. Any free inverse semigroup is F -inverse. The above composition of partial transformations of a set gives rise to a symmetric inverse semigroup. Under this alternative composition, the collection of all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in the sense of category theory. This close connection between inverse semigroups and inductive groupoids is embodied in the Ehresmann-Schein-Nambooripad Theorem , which states that an inductive groupoid can always be constructed from an inverse semigroup, and conversely.

As noted above, an inverse semigroup S can be defined by the conditions 1 S is a regular semigroup , and 2 the idempotents in S commute; this has led to two distinct classes of generalisations of an inverse semigroup: semigroups in which 1 holds, but 2 does not, and vice versa. Examples of regular generalisations of an inverse semigroup are: [32]. The class of generalised inverse semigroups is the intersection of the class of locally inverse semigroups and the class of orthodox semigroups.

Amongst the non-regular generalisations of an inverse semigroup are: [34]. This notion of inverse also readily generalizes to categories.

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An inverse category is selfdual. The category of sets and partial bijections is the prime example. Inverse categories have found various applications in theoretical computer science. There are a number of equivalent characterisations of an inverse semigroup S : [10] 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 -class and every -class contains precisely one idempotent , where and are two of Green's relations. There is therefore a simple characterisation of Green's relations in an inverse semigroup: [11] Unless stated otherwise, E S will denote the semilattice of idempotents of an inverse semigroup S. Examples of inverse semigroups Every group is an inverse semigroup.

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Every semilattice is inverse. The Brandt semigroup is inverse. The Munn semigroup is inverse. Equivalently, for some in general, different idempotent f in S. A McAlister triple is used to define the following: together with multiplication.

Lawson to: Theorem. Free inverse semigroups A construction similar to a free group is possible for inverse semigroups.

## Inverse elements

A presentation of the free inverse semigroup on a set X may be obtained by considering the free semigroup with involution , where involution is the taking of the inverse, and then taking the quotient by the Vagner congruence The word problem for free inverse semigroups is much more intricate than that of free groups.

Generalisations of inverse semigroups As noted above, an inverse semigroup S can be defined by the conditions 1 S is a regular semigroup , and 2 the idempotents in S commute; this has led to two distinct classes of generalisations of an inverse semigroup: semigroups in which 1 holds, but 2 does not, and vice versa. 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. Left, right, two-sided ample semigroups.

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

Inverse category This notion of inverse also readily generalizes to categories. CRC Press. Semigroups: An Introduction to the Structure Theory. Schein showed in that every inverse semigroup is isomorphic to an inverse semigroup of full difunctional relations and proposed the following question: given an inverse semigroup S, can we describe all of its representations by full difunctional relations? We demonstrate that each such representation may be constructed using only S itself.

It so happens that the full difunctional relations on a set X are essentially the bijections among its quotients. This observation invites us to consider Schein's question as fundamentally a problem of symmetry, as we explain.

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By Cayley's Theorem, groups are naturally represented by permutations, and more generally, every permutation representation of a group can be constructed using representations induced by its subgroups. Analogously, by the Wagner-Preston Theorem, inverse semigroups are naturally represented by one-to-one partial mappings, and every representation of an inverse semigroup can be constructed using representations induced by certain of its inverse subsemigroups.

From a universal algebraic point of view the permutations and one-to-one partial functions on a set X are the automorphisms global symmetries of X and the isomorphisms among subsets local symmetries of X, respectively. Inspired by the interpretation of difunctional relations as isomorphisms among quotients, or colocal symmetries, we introduce a class of partial algebras which we call inverse magmoids. We then show that these algebras include all inverse semigroups and groupoids and play a role among difunctional relations analogous to that played by groups among permutations and of inverse semigroups among one-to-one partial functions.