Clifford semi-group

completely-regular semi-group

A semi-group in which every element is a group element, that is, lies in some subgroup. An element of a semi-group is a group element if and only if it is completely regular (see Regular element). A semi-group $ S $ is a Clifford semi-group if and only if either of the following conditions holds: 1) For every $ a \in S $, $ a \in a ^ {2} S \cap Sa ^ {2} $; or 2) each one-sided ideal $ I $ of $ S $ is isolated (or semi-prime), that is, if $ x \notin I $ then $ x ^ {n} \notin I $ for all natural numbers $ n $.

Together with inverse semi-groups (cf. Inversion semi-group), Clifford semi-groups represent one of the most important types of regular semi-groups. Their study was begun in the fundamental paper [1] of A.H. Clifford. Every Clifford semi-group has a (unique) decomposition into groups, the classes of which are exactly the $ {\mathcal H} $- classes (cf. Green equivalence relations). Such a decomposition is not necessarily a band (see Band of semi-groups); conditions for this to be so are known (see [3]). The Green relations $ {\mathcal J} $ and $ {\mathcal D} $ on a Clifford semi-group coincide. Every completely-simple semi-group is a Clifford semi-group; a Clifford semi-group is completely simple if and only if it is simple (see Simple semi-group). Every Clifford semi-group $ S $ can be decomposed into a semi-lattice of completely-simple semi-groups; this decomposition is unique, its components are just the $ {\mathcal D} $- classes, and the corresponding quotient semi-lattice is isomorphic to the semi-lattice of principal ideals of $ S $. Conversely, every semi-group decomposable into a semi-lattice of completely-simple semi-groups is a Clifford semi-group.

The following conditions are equivalent for a Clifford semi-group: 1) $ S $ is inverse; 2) every idempotent of $ S $ lies in the centre, that is, it commutes with every element of $ S $; 3) every one-sided ideal of $ S $ is two-sided; 4) the Green relations $ {\mathcal H} $ and $ {\mathcal D} $ on $ S $ coincide; 5) $ S $ is a semi-lattice of groups; and 6) $ S $ is a subdirect product of groups and groups with zero.

The decomposition of an arbitrary Clifford semi-group into a semi-lattice of completely-simple semi-groups defines its "global structure" . The multiplication law for elements within the components of this decomposition is given by Rees' theorem (see Completely-simple semi-group). Further investigations into Clifford semi-groups are to a large extent directed towards a clarification of their "fine structure" , that is, to determining multiplication laws of elements from different components. When the components are groups, that is, for inverse Clifford semi-groups, there is a constructive description in terms of a so-called sum of a direct spectrum of groups. Let $ \{ G _ \alpha \} _ {\alpha \in A } $ be a family of pairwise disjoint groups, let $ A $ be a semi-lattice (see Idempotents, semi-group of) such that for every pair of elements $ \alpha , \beta \in A $ such that $ \alpha \geq \beta $, there is a homomorphism $ \phi _ {\alpha , \beta } : G _ \alpha \rightarrow G _ \beta $ such that $ \phi _ {\alpha , \alpha } $ is the identity automorphism for every $ \alpha $ and $ \phi _ {\alpha , \beta } \circ \phi _ {\beta , \gamma } = \phi _ {\alpha , \gamma } $ whenever $ \alpha \geq \beta \geq \gamma $. A product $ \cdot $ is defined on the union $ S = \cup _ {\alpha \in A } G _ \alpha $ by setting $ a \cdot b = a \phi _ {\alpha , \alpha \beta } b \phi _ {\beta , \alpha \beta } $ for arbitrary $ a \in G _ \alpha $ and $ b \in G _ \beta $.

Then $ S $ becomes an inverse Clifford semi-group. Conversely, every inverse Clifford semi-group can be obtained in this way.

The problem of the "fine structure" of Clifford semi-groups is in general extremely complicated, and to date (1987) there is no satisfactory solution to it. Certain very complex constructions, describing Clifford semi-groups in terms of completely-simple semi-groups, their translation hulls, semi-lattices, and mappings with special properties, are to be found in [5]. Great progress has been achieved in the case of orthodox Clifford semi-groups (see Regular semi-group); such semi-groups are called orthogroups. There are a few clear, if somewhat cumbersome, constructions for them (see [2]). All the constructions mentioned generalize, in some way, the description of inverse Clifford semi-groups obtained in [1].

References

[1]	A.H. Clifford, "Semigroups admitting relative inverses" Ann. of Math. , 42 : 4 (1941) pp. 1037–1049
[2]	A.H. Clifford, "A structure theorem for orthogroups" J. Pure Appl. Algebra , 8 : 1 (1976) pp. 23–50
[3]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[4]	E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian)
[5]	M. Petrich, "The structure of completely regular semigroups" Trans. Amer. Math. Soc. , 189 (1974) pp. 211–236

Comments

The function symbol is written after the arguments in the article above. This is common practice in the theory of semi-groups.

An extensive bibliography on recent work concerning Clifford semi-groups can be found in [a1] and in the paper of J. Meakin and K.S.S. Nambooripad in [a2].

References

[a1]	M. Petrich, "Inverse semigroups" , Wiley (1984)
[a2]	G. Pollák (ed.) St. Schwartz (ed.) O. Steinfeld (ed.) , Semigroups , Coll. Math. Soc. J. Bolyai , 39 , North-Holland (1985)