Orthodox semigroup
In mathematics, an orthodox semigroup is a regular semigroup whose set of idempotents forms a subsemigroup. In more recent terminology, an orthodox semigroup is a regular E-semigroup. The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969. Certain special classes of orthodox semigroups had been studied earlier. For example, semigroups that are also unions of groups, in which the sets of idempotents form subsemigroups were studied by P. H. H. Fantham in 1960.
Examples
- Consider the binary operation on the set S = defined by the following Cayley table :
| a | b | c | x | |
| a | a | b | c | x |
| b | b | b | b | b |
| c | c | c | c | c |
| x | x | c | b | a |
- Inverse semigroups and bands are examples of orthodox semigroups.
Some elementary properties
The set of idempotents in an orthodox semigroup has several interesting properties. Let S be a regular semigroup and for any a in S let V denote the set of inverses of a. Then the following are equivalent:- S is orthodox.
- If a and b are in S and if x is in V and y is in V then yx is in V.
- If e is an idempotent in S then every inverse of e is also an idempotent.
- For every a, b in S, if V ∩ V ≠ ∅ then V = V.