By Jorge Almeida, Benjamin Steinberg (auth.), Jean-Camille Birget, Stuart Margolis, John Meakin, Mark Sapir (eds.)

This quantity includes papers that are established totally on talks given at an inter nationwide convention on Algorithmic difficulties in teams and Semigroups held on the collage of Nebraska-Lincoln from may perhaps ll-May sixteen, 1998. The convention coincided with the Centennial social gathering of the dep. of arithmetic and facts on the collage of Nebraska-Lincoln at the celebration of the only hun dredth anniversary of the granting of the 1st Ph.D. by way of the dep.. investment used to be supplied through the U.S. nationwide technological know-how origin, the dep. of Math ematics and facts, and the school of Arts and Sciences on the collage of Nebraska-Lincoln, during the College's concentration software in Discrete, Experimental and utilized arithmetic. the aim of the convention used to be to assemble researchers with pursuits in algorithmic difficulties in team thought, semigroup idea and laptop technology. a very important characteristic of this convention was once that it supplied a framework for alternate of principles among the learn groups in semigroup thought and staff conception, and a number of other of the papers accrued the following mirror this interac tion of rules. The papers gathered during this quantity signify a go component to a number of the effects and ideas that have been mentioned within the convention. They replicate a synthesis of overlapping rules and strategies prompted through difficulties bearing on finite monoids, finitely offered mono ids, finitely offered teams and loose groups.

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**Extra info for Algorithmic Problems in Groups and Semigroups**

**Sample text**

If w = v then gO ... g, = ho ... h, in G and gO Yl . gogl Y2 ..... go .. ·g,-I Y, = ho ZI . hohl Z2 ..... ho .. ·h'-I z, (3) in F. All the elements go .. h; Zi+l belong to X, the free generators of F. By assumption, F does not satisfy any identity u ~ u' with lui, lu'l < n. Consequently, all corresponding letters must coincide, that is, go .. ·g; Yi+l = ho ... h; Zi+1 all i. Denote by wy, the product in S of the y,-values of the factors of the product (1) and by vy, the analogue for the product (2).

As is the custom in topology, a closed set Y is termed irreducible if Y = Yl U Y2, where Yl and Y2 are closed sets, implies that either Y = Y1 or Y = Y2. So, by the remark above, every closed subset Y of HIl can be expressed as a finite union of irreducible algebraic sets: Y=YIU···UYIl . These sets are usually referred to as the irreducible components of Y, which turn out to be unique. This shifts the study of algebraic sets to their irreducible components. It turns out that the irreducible ideals are the algebraic counterpart to the irreducible algebraic sets.

The following example shows that this is not possible in general. Let S be the free semigroup on two generators a, b in the variety [x 3 :::::: 0 :::::: Xl ••• X7]. Then S is a finite nilpotent semigroup contained in [x 3 :::::: x3+W] but S does not divide any direct product A x G where A E ACorn n [x 3 :::::: x 3+w ] and G e G. Suppose by contrast that S -< A x G for some G E G and A E ACom n [x 3 :::::: x 3+w ]. Let T be a subsemigroup of Ax G and let