Regular semi-group

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 [2].

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 [5]). 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 [7]), 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 [3], [5], [8], [10]). 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 [10]. 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, [5]). 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 [4], [9]).

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 [6] (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 [11] and natural [12] 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 [1], [13]). See also Semi-group.

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2]	W.D. Munn, "Regular -semigroups" Glasgow Math. J. , 9 : 1 (1968) pp. 46–66
[3]	A. Clifford, "The fundamental representation of a regular semigroup" Semigroup Forum , 10 (1975) pp. 84–92
[4]	A. Clifford, "A structure theorem for orthogroups" J. Pure Appl. Algebra , 8 (1976) pp. 23–50
[5]	K.S.S. Nambooripad, "Structure of regular semigroups, I" Mem. Amer. Math. Soc. , 22 : 224 (1979)
[6]	K.S.S. Nambooripad, "The natural partial order on a regular semigroup" Proc. Edinburgh Math. Soc. , 23 : 3 (1980) pp. 249–260
[7]	K.S.S. Nambooripad, A.R. Rajan, "Structure of combinatorial regular semigroups" Quart. J. Math. , 29 : 116 (1978) pp. 489–504
[8]	P.A. Grillet, "The structure of regular semigroups, I-IV" Semigroup Forum , 8 (1974) pp. 177–183; 254–265; 368–373
[9]	T.E. Hall, "Orthodox semigroups" Pacific. J. Math. , 39 (1971) pp. 677–686
[10]	T.E. Hall, "On regular semigroups" J. of Algebra , 24 (1973) pp. 1–24
[11]	J. Meakin, K.S.S. Nambooripad, "Coextensions of pseudo-inverse semigroups by rectangular bands" J. Austral. Math. Soc. , 30 (1980/81) pp. 73–86
[12]	R.J. Warne, "Natural regular semigroups" G. Pollák (ed.) , Algebraic Theory of Semigroups , North-Holland (1979) pp. 685–720
[13]	G. Lallement, "Structure theorems for regular semigroups" Semigroup Forum , 4 (1972) pp. 95–123