In mathematics, in semigroup theory, a Rees factor semigroup (also called Rees quotient semigroup or just Rees factor), named after David Rees, is a certain semigroup constructed using a semigroup and an ideal of the semigroup. Let S be a semigroup and I be an ideal of S. Using S and I one can construct a new semigroup by collapsing I into a single element while the elements of S outside of I retain their identity. The new semigroup obtained in this way is called the Rees factor semigroup of S modulo I and is denoted by S/I. The concept of Rees factor semigroup was introduced by David Rees in 1940.

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, and similarly for IS). Let I be an ideal of a semigroup S. The relation \rho in S defined by 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, where 0 is a new element and the product (here denoted by 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 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.