Narrow Results

We construct the inverse partition semigroup, isomorphic to the dual symmetric inverse monoid, introduced in [6]. We give a convenient geometric illustration for elements of . We describe all maximal subsemigroups of and find a generating set for when X is finite. We prove that all the automorphisms of are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets X, we establish that, up to equivalence, there is a unique faithful effective transitive representation of , namely to . Finally, we construct an interesting -cross-section of , which is reminiscent of , the -cross-section of , constructed in [4].

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.

We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a $p$-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic presentation of these finite solvable groups.