Regular semigroup

A regular semigroup is a semigroup "S" in which every element is regular, i.e., for each element "a", there exists an element "x" such that "axa" = "a". [Howie 1995 : 54.] Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations. [Howie 2002.]

Origins

Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of "regularity" in a semigroup was adapted from an analogous condition for rings, already considered by J. von Neumann. [von Neumann 1936.] It was his study of regular semigroups which led Green to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees.

The basics

There are two equivalent ways in which to define a regular semigroup "S"::(1) for each "a" in "S", there is an "x" in "S", which is called a pseudoinverse [Klip, Knauer and Mikhalev : p. 33] , with "axa" = "a";:(2) every element "a" has at least one inverse "b", in the sense that "aba" = "a" and "bab" = "b".To see the equivalence of these definitions, first suppose that "S" is defined by (2). Then "b" serves as the required "x" in (1). Conversely, if "S" is defined by (1), then "xax" is an inverse for "a", since "a"("xax")"a" = "axa"("xa") = "axa" = "a" and ("xax")"a"("xax") = "x"("axa")("xax") = "x"("axa")"x" = "xax". [Clifford and Preston 1961 : Lemma 1.14.]

The set of inverses (in the above sense) of an element "a" in an arbitrary semigroup "S" is denoted by "V"("a"). [Howie 1995 : p. 52.] Thus, another way of expressing definition (2) above is to say that in a regular semigroup, "V"("a") is nonempty, for every "a" in "S". The product of any element "a" with any "b" in "V"("a") is always idempotent: "abab" = "ab", since "aba" = "a". [Clifford and Preston 1961 : p. 26.]

A regular semigroup in which idempotents commute is an inverse semigroup, that is, every element has a "unique" inverse. To see this, let "S" be a regular semigroup in which idempotents commute. Then every element of "S" has at least one inverse. Suppose that "a" in "S" has two inverses "b" and "c", i.e.,:"aba" = "a", "bab" = "b", "aca" = "a" and "cac" = "c".Then:"b" = "bab" = "b"("aca")b" = "bac"("ac")("ab") = "bac"("ab")("ac") = ("ca")("ba")"bac" = "cabac" = "cac" = "c".So, by commuting the pairs of idempotents "ab" & "ac" and "ba" & "ca", the inverse of "a" is shown to be unique. Conversely, it can be shown that any inverse semigroup is a regular semigroup in which idempotents commute. [Howie 1995 : Theorem 5.1.1.]

The existence of a unique pseudoinverse implies the existence of a unique inverse, but the opposite is not true. For example, in the symmetric inverse semigroup, the empty transformation Ø does not have a unique pseudoinverse, because Ø = Ø"f"Ø for any transformation "f". The inverse of Ø is unique however, because only one "f" satisfies the additional constraint that "f" = Ø"f"Ø, namely "f" = Ø. This remark holds more generally in any semigroup with zero. Furthermore, if any element has a unique pseudoinverse, then the semigroup is a group, and the unique pseudoinverse of an element coincides with the group inverse [Proof: http://planetmath.org/?op=getobj&from=objects&id=6391] .

Theorem. The homomorphic image of a regular semigroup is regular. [Howie 1995 : Lemma 2.4.4.]

Examples of regular semigroups:
*Every group is regular.
*Every inverse semigroup is regular.
*Every band is regular.
*The bicyclic semigroup is regular.
*Any full transformation semigroup is regular.
*A Rees matrix semigroup is regular.

Green's Relations

Recall that the principal ideals of a semigroup "S" are defined in terms of "S"¹, the "semigroup with identity adjoined"; this is to ensure that an element "a" belongs to the principal right, left and two-sided ideals which it generates. In a regular semigroup "S", however, an element "a" = "axa" automatically belongs to these ideals, without recourse to adjoining an identity. Green's relations can therefore be redefined for regular semigroups as follows::$a,mathcal\{L\},b$ if, and only if, "Sa" = "Sb";:$a,mathcal\{R\},b$ if, and only if, "aS" = "bS";:$a,mathcal\{J\},b$ if, and only if, "SaS" = "SbS". [Howie 1995 : 55.]

In a regular semigroup "S", every $mathcal\{L\}$- and $mathcal\{R\}$-class contains at least one idempotent. If "a" is any element of "S" and α is any inverse for "a", then "a" is $mathcal\{L\}$-related to "αa" and $mathcal\{R\}$-related to "aα". [Clifford and Preston 1961 : Lemma 1.13.]

Theorem. Let "S" be a regular semigroup, and let "a" and "b" be elements of "S". Then
*$a,mathcal\{L\},b$ if, and only if, there exist α in "V"("a") and β in "V"("b") such that α"a" = β"b";
*$a,mathcal\{R\},b$ if, and only if, there exist α in "V"("a") and β in "V"("b") such that "a"α = "b"β. [Howie 1995 : Proposition 2.4.1.]

If "S" is an inverse semigroup, then the idempotent in each $mathcal\{L\}$- and $mathcal\{R\}$-class is unique. [Howie 1995 : Theorem 5.1.1.]

Special classes of regular semigroups

Some special classes of regular semigroups are: [Howie 1995 : Section 2.4 & Chapter 6.]
*"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 of locally inverse semigroups and the class of orthodox semigroups. [Howie 1995 : 222.]

Notes

References

*A. H. Clifford and G. B. Preston, "The Algebraic Theory of Semigroups", Volume 1, Mathematical Surveys of the American Mathematical Society, No. 7, Providence, R.I., 1961.
*J. M. Howie, "Fundamentals of Semigroup Theory", Clarendon Press, Oxford, 1995.
*M. Kilp, U. Knauer, A.V. Mikhalev, "Monoids, Acts and Categories with Applications to Wreath Products and Graphs", De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3110152487.
*cite journal | author=J. A. Green | title=On the structure of semigroups | journal=Annals of Mathematics (2) | year=1951 | volume=54 | pages=163–172 | doi=10.2307/1969317
*J. M. Howie, Semigroups, past, present and future, "Proceedings of the International Conference on Algebra and Its Applications", 2002, 6-20.
*cite journal | author=J. von Neumann | title=On regular rings | journal=Proceedings of the National Academy of Sciences of the USA | year=1936 | volume=22 | pages=707–713 | doi=10.1073/pnas.22.12.707