# 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