[Paper Review] Inverse semigroups with rational word problem are finite

Tara Brough|arXiv (Cornell University)|Nov 15, 2013

semigroups and automata theory10 references3 citations

TL;DR

This paper proves that inverse semigroups with a rational word problem—defined as the set of pairs of words representing the same element being recognizable by a two-tape asynchronous finite automaton—are necessarily finite. The proof uses structural analysis of monogenic inverse semigroups and shows that the monogenic free inverse semigroup, the only infinite candidate, cannot have a rational word problem due to a pumping lemma-style contradiction in word equivalence.

ABSTRACT

This note proves a generalisation to inverse semigroups of Anisimov's theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word problem -- the set of all pairs of words over a generating set for the semigroup which both represent the same element.

Motivation & Objective

• To resolve a question posed by Stuart Margolis on whether infinite inverse semigroups can have a rational word problem.
• To extend Anisimov’s theorem—stating that a group has a regular word problem iff it is finite—to the class of inverse semigroups.
• To show that the only inverse semigroups with rational word problem are finite, by analyzing the structure of monogenic inverse semigroups.
• To demonstrate that the monogenic free inverse semigroup, the only possible infinite candidate, does not have a rational word problem.
• To establish that rational word problem in inverse semigroups implies finiteness, using automata-theoretic and semigroup-theoretic techniques.

Proposed method

• Define the two-tape word problem as the relation of word pairs over a generating set that represent the same element in the semigroup.
• Use asynchronous finite state automata (AFSA) to formalize the notion of a rational word problem, where acceptance depends on parallel, non-simultaneous reading of two input tapes.
• Apply known results: rational word problem is independent of generating set choice, and finitely generated subsemigroups of rational word problem semigroups are also in the class.
• Analyze the classification of monogenic inverse semigroups into types: periodic, infinite cyclic, bicyclic, or free monogenic inverse semigroup.
• Use a pumping argument on an AFSA recognizing the word problem of the free monogenic inverse semigroup, showing it would accept invalid word pairs.
• Contradict the automaton’s acceptance by computing in a standard model of the free inverse semigroup, proving that certain word pairs are not equal despite being accepted.

Experimental results

Research questions

• RQ1 Can there exist an infinite inverse semigroup with a rational word problem?
• RQ2 Does the rational word problem condition in inverse semigroups imply finiteness, as it does in groups?
• RQ3 Is the monogenic free inverse semigroup, the only infinite inverse semigroup that could potentially have a rational word problem, actually excluded by the rationality condition?
• RQ4 Can the structure of inverse semigroups with rational word problem be fully characterized, and does it exclude all infinite examples?
• RQ5 How does the two-tape automaton model constrain the equality of words in inverse semigroups, particularly in non-free settings?

Key findings

• The only inverse semigroups with a rational word problem are finite, generalizing Anisimov’s theorem to inverse semigroups.
• The monogenic free inverse semigroup does not have a rational word problem, as shown by a contradiction arising from automaton pumping and word equality in a standard model.
• An asynchronous finite automaton recognizing the word problem of the free monogenic inverse semigroup would accept a pair of words that are not equal in the semigroup, contradicting correctness.
• The proof relies on the fact that infinite subsemigroups of rational word problem semigroups must contain elements of infinite order, and such elements in inverse semigroups lead to the free monogenic inverse semigroup.
• The absence of the bicyclic monoid and infinite subgroups in rational word problem semigroups further restricts the possible infinite structures in inverse semigroups.
• The result confirms that rational word problem is a strong finiteness condition in the class of inverse semigroups, analogous to regularity in group word problems.