Rees factor semigroup

Formal definition

A subset I of a semigroup S is called an ideal of S if both SI and IS are subsets of I (where SI = {sx \mid s \in S \text{ and } x \in I}, and similarly for IS). Let I be an ideal of a semigroup S. The relation \rho in S defined by

: x ρ y ⇔ either x = y or both x and y are in I

is an equivalence relation in S. The equivalence classes under \rho are the singleton sets {x} with x not in I and the set I. Since I is an ideal of S, the relation \rho is a congruence on S. The quotient semigroup S/{\rho} is, by definition, the Rees factor semigroup of S modulo I. For notational convenience the semigroup S/\rho is also denoted as S/I. The Rees factor semigroup has underlying set (S \setminus I) \cup {0}, where 0 is a new element and the product (here denoted by *) is defined by

s * t = \begin{cases} st & \text{if } s, t, st \in S \setminus I \ 0 & \text{otherwise}. \end{cases}

The congruence \rho on S as defined above is called the Rees congruence on S modulo I.

Example

Consider the semigroup S = { a, b, c, d, e } with the binary operation defined by the following Cayley table:

Let I = { a, d } which is a subset of S. Since

:SI = { aa, ba, ca, da, ea, ad, bd, cd, dd, ed } = { a, d } ⊆ I :IS = { aa, da, ab, db, ac, dc, ad, dd, ae, de } = { a, d } ⊆ I

the set I is an ideal of S. The Rees factor semigroup of S modulo I is the set S/I = { b, c, e, I } with the binary operation defined by the following Cayley table:

Ideal extension

A semigroup S is called an ideal extension of a semigroup A by a semigroup B if A is an ideal of S and the Rees factor semigroup S/A is isomorphic to B.

Some of the cases that have been studied extensively include: ideal extensions of completely simple semigroups, of a group by a completely 0-simple semigroup, of a commutative semigroup with cancellation by a group with added zero. In general, the problem of describing all ideal extensions of a semigroup is still open.

References

References

1. (1940). "On semigroups". [[Mathematical Proceedings of the Cambridge Philosophical Society]].
2. (1961). "The algebraic theory of semigroups. Vol. I". [[American Mathematical Society]].
3. Lawson (1998) ''Inverse Semigroups: the theory of partial symmetries'', page 60, [[World Scientific]] with [https://books.google.com/books?id=2805q4tFiCkC&pg=PA60 Google Books link]
4. Howie, John M.. (1995). "Fundamentals of Semigroup Theory". [[Clarendon Press]].
5. "The concise handbook of algebra". [[Springer Science+Business Media. Springer]]. (2002 (pp. 1–3))
6. Gluskin, L.M.. "Extension of a semi-group".