
Inverse elements
a=axa, x=xax
In this page I give some results I have found with my big interest on the idea of inverse elements and related ideas of "inverse" in regular semigroups.
 Some attempts to formally generalize inverse semigroups
[last changed April 21, 2004; February 5, 2007.] Estonian version is exactly my master thesis, but English version is a bit shortened, repaired according to oponents remarks, and updated with the answer to the open question in my master thesis. It is also present in ResearchGate.
[PDF] in Estonian / [PDF] in English /
[DVI] in Estonian / [DVI] in English /
[PS] in Estonian / [PS] in English /
Abstract
In this paper I give results I have found when I tried to generalize
inverse semigroups. Instead two new descriptions for semilattices, one for
groups, one for Einversive semigroups and one for completely simple semigroups is found.
Details
The results in this file are taken from my bachelor thesis and master thesis. In the sequel if not stated
otherwise, k is a positive integer.
§1. First there is a short introduction to inverse and regular semigroups
with definitions and one basic result.
§2. I introduce new concepts kregular semigroup (every element
s has kpseudoinverse elements: s=ss_{1}s_{2}...s_{k}s) and kinverse semigroup (every element has unique kinverse elements).
Then inverse semigroup is precisely 1inverse and seemingly we have something more general. The basic idea could be seen from
the following definiton:
Results
I.2.1. S is regular iff it is kregular.
I.2.2. S is regular iff every element has kinverse elements.
I.2.3. If k>=2, then S is kinverse iff it is semilattice.
§3. I introduce a new concept weakly kinverse semigoup (the uniqueness
is up to the set of kinverse elements, not piecewise). Then kinverse semigroup
is weakly kinverse, inverse semigroup is precisely weakly 1inverse. Seemingly we have again something generalized.
Results
I.3.1. If k>=2, then S is weakly kinverse iff it is semilattice.
§4. I introduce new concepts kturnregular (every k elements have an element, for
which they are kpseudoinverse), almost kturninverse (every k elements have an element, for which they are
kinverse) and kturninverse (the same, but that element is unique) semigroups.
Then kturninverse semigroup is almost kturninverse, which is kturnregular.
Results
I.4.1. S is kturnregular iff it is Einversive.
I.4.2. S is almost 1turninverse iff it is regular.
I.4.3. S is 1turninverse iff it is inverse.
I.4.4. If k>=2, then S is kturninverse iff it is group.
I.4.5. If k>=2, then S is almost kturninverse iff it is completely simple.
References
[Ho] Howie, J. M., "Fundamentals of Semigroup Theory", Clarendon Press, Oxford, 1995.
[Ki] Kilp, M., "Algebra I", Tartu, 1998.
[La] Lawson, M. V., "Inverse Semigroups  The Theory of Partial Symmetries", World Scientific, 1998.
[Mi] Mitsch, H., Subdirect products of Einversive semigroups, J. Austral. Math. Soc. (Series A),
48 (1990), 6678.
[Mu] Must, R., kinverse semigroups and the construction of onedimensional tiling semigroups,
bachelor thesis, Tartu, 2002.
[Pe] Petrich, M., Reilly, Norman R., "Completely Regular Semigroups", John Wiley & Sons, Inc., 1999.
[We] Weipoltshammer, B., Certain Congruences on Einversive Esemigroups, Semigroup Forum, 65 (2002), 233248.
 Ideas and questions
[last changed January 16, 2004]
These ideas and questions are short, so I give pictures:
Idea 01  Generalize regularity and Einversivity with one definiton (10.2003)
Idea 02  Certain subsets of a semigroup (10.2003)
Idea 03  Simple question? (10.2003)
Idea 04  1/3 and 2/3 associative dense groupoids (15.01.2004)
Idea 03 is actually a good exercise for a mathinterested student.
 Contact
Riivo Must, MSc
