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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.J. Almeida, J.-É. Pin and P. Weil [http://hal.archives-ouvertes.fr/docs/00/06/02/12/PDF/Cambridge.pdf Semigroups whose idempotents form a subsemigroup] updated version of {{Cite journal | doi = 10.1017/S0305004100075332| title = Semigroups whose idempotents form a subsemigroup| journal = Mathematical Proceedings of the Cambridge Philosophical Society| volume = 111| issue = 2| pages = 241| year = 2008| last1 = Almeida | first1 = J.| last2 = Pin | first2 = J.-É. | last3 = Weil | first3 = P.| s2cid = 6344747| url = https://hal.archives-ouvertes.fr/hal-00019881/document}} The term orthodox semigroup was coined by T. E. Hall and presented in a paper published in 1969.{{cite journal|title=On regular semigroups whose idempotents form a subsemigroup|journal=Bulletin of the Australian Mathematical Society|date=1969|volume=1|pages=195–208|doi=10.1017/s0004972700041447 | last1 = Hall | first1 = T. E.|issue=2 |doi-access=free}}{{cite book|editor1-first=A. H.|editor1-last=Clifford|editor2-first=K. H.|editor2-last=Hofmann|editor3-first=M. W.|editor3-last=Mislove|title=Semigroup Theory and Its Applications: Proceedings of the 1994 Conference Commemorating the Work of Alfred H. Clifford|date=1996|publisher=Cambridge University Press|isbn=9780521576697|page=70}} 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.{{cite journal|last1=P.H.H. Fantham|title=On the Classification of a Certain Type of Semigroup|journal=Proceedings of the London Mathematical Society |date=1960|volume=1|pages=409–427|doi=10.1112/plms/s3-10.1.409}}

Examples

class="wikitable" style="margin:1em auto;"
         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 }.

  • Inverse semigroups and bands are examples of orthodox semigroups.{{cite book|last1=P.A. Grillet|title=Semigroups: An introduction to structure theory|publisher=Marcel Dekker, Inc.|location=New York|page=341}}

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:{{cite book|last1=J.M. Howie|title=An introduction to semigroup theory|date=1976|publisher=Academic Press|location=London|pages=186–211}}

  • 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).

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.

See also

References

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Category:Semigroup theory