# Regular semi-group

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A semi-group in which every element is regular (see Regular element).

An arbitrary regular semi-group $S$ contains idempotents (see Idempotent), and the structure of $S$ is determined to a considerable extent by the "structure" and the "distribution" in $S$ of the set $E ( S)$ of all its idempotents (cf. Idempotents, semi-group of). Regular semi-groups with a unique idempotent are just groups. In the first place, $E ( S)$ can be regarded as a partially ordered set in a natural way. There are known structure theorems describing a regular semi-group $S$ with certain natural restrictions on the set $E ( S)$. One such restriction (for semi-groups with zero) is that all non-zero idempotents are primitive (see Completely-simple semi-group); a semi-group with this property is called primitive. The following conditions on a semi-group $S$ are equivalent: a) $S$ is a primitive regular semi-group; b) $S$ is a regular semi-group equal to the union of its $O$- minimal (right) ideals (see Minimal ideal); and c) $S$ is an $O$- direct union of completely $0$- simple semi-groups. The structure of regular semi-groups is also known in the case when $E ( S)$ is a chain with the order type of the negative integers .

A more informative view of $E ( S)$ is obtained if one defines a partial operation $\circ$ on it in the following way. If $e , f \in E ( S)$ are such that at least one of the products $e f$, $f e$ is equal to either $e$ or $f$, then $e f \in E ( S)$; one then sets $e \circ f = e f$. The resulting partial algebra can be axiomatized in terms of two quasi-order relations $\omega ^ {r}$ and $\omega ^ {l}$. These are closely related to the given partial operation (the realization of these relations in $E ( S)$ is as follows: $e \omega ^ {r} f$ means $f e = e$, $e \omega ^ {l} f$ means $e f = e$; then $\omega ^ {r} \cap \omega ^ {l}$ is the natural partial order on $E ( S)$). Such a partial algebra is called a bi-ordered set (see ). An arbitrary regular semi-group can be constructed in a specific way from a bi-ordered set and groups. It is thus possible to classify regular semi-groups in terms of bi-ordered sets. Among the types of semi-groups that have been investigated in this way are combinatorial regular semi-groups (see ), that is, those whose only subgroups consist of one element.

A homomorphic image of a regular semi-group is regular. Every normal complex of a regular semi-group which is a sub-semi-group contains an idempotent. An arbitrary congruence (cf. Congruence (in algebra)) on a regular semi-group is uniquely determined by its classes that contain idempotents. A congruence on a regular semi-group $S$ separates idempotents if and only if it is contained in the relation ${\mathcal H}$( see Green equivalence relations). The set of such congruences forms a modular sublattice with a zero and a unit element in the lattice of all congruences on $S$( cf. also Modular lattice). A regular semi-group is called fundamental if this sublattice contains only the equality relation. Every combinatorial regular semi-group is fundamental. Fundamental regular semi-groups are important, not only as one of the more visible types of regular semi-groups, but also because of their "universality" property in the class of all semi-groups. More precisely, for any bi-ordered set $E$ it is possible to construct in a canonical way a fundamental regular semi-group $T _ {E}$ such that $E$ is the bi-ordered set of all idempotents, and for any regular semi-group $S$ with $E ( S) = E$ there is a homomorphism $\phi : S \rightarrow T _ {E}$ that separates idempotents and is such that $\phi ( S)$ is a sub-semi-group of $T _ {E}$ containing $E$( for various constructions of $T _ {E}$, see , , , ). A regular semi-group $S$ is fundamental if and only if $\phi$ is injective.

If $S$ is a regular semi-group, then the sub-semi-group $\langle E ( S) \rangle$ generated by its idempotents is also regular. The sub-semi-group $\langle E ( S) \rangle$ exerts an essential influence on the structure of $S$. A regular semi-group is idempotently generated if and only if the same is true for each of its principal factors . In an idempotently-generated regular semi-group $S$, any element $x$ can be written in the form $x = e _ {1} \dots e _ {n}$, where $e _ {i} \in E ( S)$ and $e _ {i} ( {\mathcal L} \cup {\mathcal R} ) e _ {i+} 1$ for $i = 1 \dots n - 1$( here ${\mathcal L}$ and ${\mathcal R}$ are Green equivalence relations, ). A sequence of idempotents $e _ {1} \dots e _ {n}$ with the above property is called an $E$- chain. In a bi-simple idempotently-generated semi-group, any two idempotents are connected by an $E$- chain, and if they are comparable in the sense of the natural partial order, then such a chain has length $\geq 4$.

If $\langle E ( S) \rangle = E( S)$, that is, the product of any two idempotents is again an idempotent, then the regular semi-group $S$ is called orthodox. The class of orthodox semi-groups contains, in particular, all inverse semi-groups. A semi-group is orthodox if and only if its principal factors are. There are structure theorems for orthodox semi-groups (see , ).

The natural partial order on $E ( S)$ can be extended to the regular semi-group $S$ in the following way: $x \leq y$ if there are idempotents $e$ and $f$ such that $x = e y = y f$. If $S$ is inverse, the relation $\leq$ becomes the natural partial order, and it is also called the natural partial order for an arbitrary regular semi-group. The relation $\leq$ on the regular semi-group $S$ is compatible with the multiplication if and only if, for any idempotent $e$, the sub-semi-group $e S e$ is inverse  (cf. Inversion semi-group). Regular semi-groups with this property are called pseudo-inverse. A wider class is formed by pseudo-orthodox semi-groups (those in which the sub-semi-group $e S e$ is orthodox for any idempotent $e$). These classes of semi-groups are also called "locally inverse regular semi-grouplocally inverse" and "locally orthodox regular semi-grouplocally orthodox" , respectively. A regular semi-group is called natural if the set of all its group elements (see Regular element) is a sub-semi-group. There are structure theorems for pseudo-inverse, pseudo-orthodox  and natural  regular semi-groups.

Numerous structure theorems for various types of regular semi-groups represent (sometimes very remote) generalizations and modifications of the structure of a Rees semi-group of matrix type or of the sum of the direct spectrum of groups (see Clifford semi-group), and are based on various representations of semi-groups and their decomposition into subdirect products (see , ). See also Semi-group.

How to Cite This Entry:
Regular semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_semi-group&oldid=48487
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article