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 $I$ of a semigroup $S$ is called an ''ideal'' of $S$ if both $SI$ and $IS$ are subsets of $I$ (where $SI = \$, 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 $\$ 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 $S/$ 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 \$, where $0$ is a new element and the product (here denoted by
$*$) is defined by

$s * t = \begin st & \text s, t, st \in S \setminus I \\ 0 & \text. \end$

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

Example

Consider the semigroup ''S'' = with the binary operation defined by the following Cayley table:

Let ''I'' = which is a subset of ''S''. Since

:''SI'' = = ⊆ ''I''
:''IS'' = = ⊆ ''I''

the set ''I'' is an ideal of ''S''. The Rees factor semigroup of ''S'' modulo ''I'' is the set ''S''/''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

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{{PlanetMath attribution, id=3517, title=Rees factor

Semigroup theory