Top Qs
Timeline
Chat
Perspective

Orthodox semigroup

From Wikipedia, the free encyclopedia

Remove ads

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.[1] The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969.[2][3] 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.[4]

Remove ads

Examples

         a    b    c    x  
  a  a  b  c  x
  b  b  b  b  b
  c  c  c  c  c
  x  x  c  b  a
Then S is an orthodox semigroup under this operation, the subsemigroup of idempotents being { a, b, c }.[5]
Remove ads

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(a) denote the set of inverses of a. Then the following are equivalent:[5]

  • S is orthodox.
  • If a and b are in S and if x is in V(a) and y is in V(b) then yx is in V(ab).
  • If e is an idempotent in S then every inverse of e is also an idempotent.
  • For every a, b in S, if V(a)  V(b) ≠ ∅ then V(a) = V(b).
Remove ads

Structure

The structure of orthodox semigroups have been determined in terms of bands and inverse semigroups. The Hall–Yamada pullback theorem describes this construction. The construction requires the concepts of pullbacks (in the category of semigroups) and Nambooripad representation of a fundamental regular semigroup.[6]

See also

References

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads