Rees factor semigroup

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 formula_1 of a semigroup formula_2 is called an "ideal" of formula_2 if both formula_4 and formula_5 are subsets of formula_1 (where formula_7, and similarly for formula_5). Let formula_1 be an ideal of a semigroup formula_2. The relation formula_11 in formula_2 defined by
is an equivalence relation in formula_2. The equivalence classes under formula_11 are the singleton sets formula_15 with formula_16 not in formula_1 and the set formula_1. Since formula_1 is an ideal of formula_2, the relation formula_11 is a congruence on formula_2. The quotient semigroup formula_23 is, by definition, the "Rees factor semigroup" of formula_2 modulo
formula_1. For notational convenience the semigroup formula_26 is also denoted as formula_27. The Rees factor
semigroup has underlying set formula_28, where formula_29 is a new element and the product (here denoted by 
formula_30) is defined by
formula_31
The congruence formula_11 on formula_2 as defined above is called the "Rees congruence" on formula_2 modulo formula_1.
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"&hairsp;/"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"&hairsp;/"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.