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.

1.   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 E-inversive 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 k-regular semigroup (every element s has k-pseudoinverse elements: s=ss1s2...sks) and k-inverse semigroup (every element has unique k-inverse elements). Then inverse semigroup is precisely 1-inverse 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 k-regular.
I.2.2. S is regular iff every element has k-inverse elements.
I.2.3. If k>=2, then S is k-inverse iff it is semilattice.

§3. I introduce a new concept weakly k-inverse semigoup (the uniqueness is up to the set of k-inverse elements, not piecewise). Then k-inverse semigroup is weakly k-inverse, inverse semigroup is precisely weakly 1-inverse. Seemingly we have again something generalized.
Results
I.3.1. If k>=2, then S is weakly k-inverse iff it is semilattice.

§4. I introduce new concepts k-turnregular (every k elements have an element, for which they are k-pseudoinverse), almost k-turninverse (every k elements have an element, for which they are k-inverse) and k-turninverse (the same, but that element is unique) semigroups. Then k-turninverse semigroup is almost k-turninverse, which is k-turnregular.
Results
I.4.1. S is k-turnregular iff it is E-inversive.
I.4.2. S is almost 1-turninverse iff it is regular.
I.4.3. S is 1-turninverse iff it is inverse.
I.4.4. If k>=2, then S is k-turninverse iff it is group.
I.4.5. If k>=2, then S is almost k-turninverse 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 E-inversive semigroups, J. Austral. Math. Soc. (Series A), 48 (1990), 66-78.
[Mu] Must, R., k-inverse semigroups and the construction of one-dimensional 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 E-inversive E-semigroups, Semigroup Forum, 65 (2002), 233-248.

2.   Ideas and questions [last changed January 16, 2004]

These ideas and questions are short, so I give pictures:
Idea 01 - Generalize regularity and E-inversivity 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 math-interested student.

3.   Contact

Riivo Must, MSc  Since 20.10.03