Rees factor semigroup

Formal definition

A subset I of a semigroup S is called an ideal of S if both S I and I S are subsets of I . Let I be an ideal of a semigroup S . The relation ρ 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 ρ are the singleton sets { x } with x not in I and the set I . Since I is an ideal of S , the relation ρ is a congruence on S . The quotient semigroup S / ρ is, by definition, the Rees factor semigroup of S modulo I . For notational convenience the semigroup S / ρ is also denoted as S / I . The Rees factor semigroup has underlying set ( S ∖ I ) ∪ { 0 } , where 0 is a new element and the product (here denoted by ∗ ) is defined by

s ∗ t = { s t if  s , t , s t ∈ S ∖ I 0 otherwise .

The congruence ρ 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 } ⊆ IIS = { 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.