Decidability and independence of conjugacy problems in finitely presented monoids

Abstract

There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by ∼p, ∼o, and ∼c) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, p-conjugacy is "almost" transitive, ∼c is strictly included in ∼p, and the p-and c-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show that there exists a monoid M defined by a finite complete presentation such that the c-conjugacy problem for M is undecidable, and that for finitely presented monoids, the c-conjugacy problem and the word problem are independent, as are the c-conjugacy and p-conjugacy problems.

On equations and first-order theory of one-relator monoids

“…This theorem extends and improves on the result (see e.g. [3]) that polycyclic monoids have decidable conjugacy problem.…”

“…Several notions of conjugacy have been investigated for monoids, and all of them are expressible in terms of the existence of solutions to certain sets of equations. Results on decidability of conjugacy problems for monoids may be found in [3,4,13,73,76,92,93]. In particular, the decidability and complexity of conjugacy problems in polycyclic monoids is investigated in [3], and in [92] Zhang proves that the conjugacy problem is decidable in all one-relator monoids of the form xA | u n " 1y with n ą 1.…”

“…Results on decidability of conjugacy problems for monoids may be found in [3,4,13,73,76,92,93]. In particular, the decidability and complexity of conjugacy problems in polycyclic monoids is investigated in [3], and in [92] Zhang proves that the conjugacy problem is decidable in all one-relator monoids of the form xA | u n " 1y with n ą 1. Two natural definitions of conjugation for monoids that have been considered in the literature are the following: x is conjugate to y if there exist u, v such that x " uv and y " vu; or x is conjugate to y if there exists w such that wx " yw.…”

On equations and first-order theory of one-relator monoids

“…We would like to emphasize how the Diophantine problem generalizes and contains many well-known and studied algorithmic problems. Notably, and as already mentioned, both the word problem and the conjugacy problem are particular cases of the Diophantine problem (see [3,4,53,54,55,69,70] for definitions and results regarding the conjugacy problem in monoids). Moreover, the left and right divisibility problems in monoids, as well as decidability of Green's orders ď R and ď L are particular instances of the Diophantine problem.…”