Representation of a semi-group

$S$ in a class of semi-groups $\Sigma$

A homomorphism of $S$ into a semi-group from $\Sigma$ (in the case of an injective homomorphism one speaks of a faithful representation). One usually has in mind a class of concrete semi-groups. Representations in a class of transformation semi-groups (for short — representations by transformations, cf. also Transformation semi-group), in a class of semi-groups of partial transformations or binary relations, or in a class of matrix semi-groups (so-called matrix, or linear, representations of a semi-group) have been studied most completely. In automata theory, with each automaton one associates a representation of the free semi-group of transformations of the set of its internal states. Representations of a semi-group by transformations that are in some way or other related to properties of the elements of the transformed set, endowed with some structure, (endomorphisms, continuous transformations, etc.) have special significance. Every semi-group with an identity element can be isomorphically represented as the semi-group of all endomorphisms of an oriented or non-oriented graph, as the semi-group of all endomorphisms of an algebra with unary operations, etc. A few constructions for obtaining all representations of a semi-group by partial transformations are known (1983). These representations are constructed from certain simple representations by union, multiple repetition, restriction to a subset, and by the operation of imbedding of semi-groups.

By adjoining an identity element to a semi-group $S$, $S^1 = S \cup \{1\}$, and by extending the regular representation of $S$ by left shifts onto a Cartesian power $S^J$ of $S^1$, one obtains a representation $\phi_J$ — the $J$-fold repetition of the regular representation of $S$. Every representation $\psi$ of $S$ by transformations of a set $\Omega$ can be obtained (cf. [2]) from $\phi_J$ by means of a mapping $\theta:S^J\rightarrow \Omega$ such that $$ \psi a(\theta\alpha) = \theta(\phi_J a(\alpha))\,\ \ a\in S\,,\ \alpha\in S^J \ . $$

Transitive representations (i.e. representations $\phi$ by transformations of a set $\Omega$ such that for any $\alpha,\beta \in \Omega$ there is an $a \in S$ for which $\phi a(\alpha) = \beta$) play a special role.

The representations of a semi-group are, by one-to-one partial transformations, related to the notion and properties of inverse semi-groups (cf. Inversion semi-group).

For the study of matrix representations of a semi-group one invokes semi-group algebras (cf. Semi-group algebra). The problem of reducibility of matrix representations has been studied. For a number of semi-groups (including finite semi-groups) the irreducible representations (cf. Irreducible representation) have been found. The matrix representations of completely-simple and completely $0$-simple semi-groups (cf. Completely-simple semi-group) can be constructed as extensions of representations of their subgroups. The matrix representations of arbitrary semi-groups can be described by means of representations of their factors, which are simple and $0$-simple semi-groups.

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2]	V.V. Vagner, "Representations of ordered semi-groups" Mat. Sb. , 38 : 2 (1956) pp. 203–240 (In Russian)
[3]	E.S. Lyapin, "Representations of semi-groups by partial mappings" Mat. Sb. , 52 : 1 (1960) pp. 589–596 (In Russian)
[4]	B.M. Shain, "Representations of semi-groups by binary relations" Mat. Sb. , 60 : 3 (1963) pp. 293–303 (In Russian)
[5a]	D.B. McAlister, "Representations of semigroups by linear transformations I" Semi-group Forum , 2 : 3 (1971) pp. 189–263
[5b]	D.B. McAlister, "Representations of semigroups by linear transformations II" Semi-group Forum , 2 : 4 (1971) pp. 283–320
[6]	B. Jónsson, "Topics in universal algebra" , Springer (1972)