# Presentations of completely regular semigroups

#### Abstract

If $\langle X\vert\kappa\sb{\rm CR}\rangle$ is a presentation of a completely regular semigroup, then the question arises whether this presentation gives rise to a solvable word problem or not. More explicitly, given any words $w,w\prime$ in the free unary semigroup U(X), is it decidable whether w and $w\prime$ represent the same element in the completely regular semigroup determined by the above presentation? If this is decidable we say that this presentation has a solvable word problem and also that the corresponding completely regular semigroup has a solvable word problem. In one presentation considered here, it is shown that the solvability of the word problem for this completely regular semigroup presentation is equivalent to the solvability of the word problem for the corresponding group presentation. Using another type of presentation it is shown that every finitely presented group is the greatest group homomorphic image of a finitely presented completely regular semigroup with a solvable word problem. Some presentations in some important subvarieties of CR, such as, O the variety of orthogroups, OBG the variety of orthodox band of groups, CS the variety of completely simple semigroups are considered. In all these cases the solution of the word problem can be reduced to the solution of the word problem some finitely many finite group presentations. Also, if an orthogroup presentation gives rise to a solvable word problem then the corresponding group presentation has a solvable word problem. An example is given to show that the converse of the above is false.

#### Recommended Citation

Ajan, Komanamana S., "Presentations of completely regular semigroups" (1993). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI9411495.

https://epublications.marquette.edu/dissertations/AAI9411495