[Paper Review] Infinite irredundant equational axiomatisability for a finite monoid

Marcel Jackson|arXiv (Cornell University)|Nov 18, 2015

Advanced Algebra and Logic17 references3 citations

TL;DR

This paper constructs a finite nilpotent monoid whose equational theory admits an infinite irredundant basis, resolving an open problem in universal algebra. By carefully analyzing variable occurrences and subword structures in equations, the author proves that certain long equations cannot be derived from shorter ones, ensuring no finite subset can serve as a complete, non-redundant axiomatization—demonstrating that infinite irredundant equational axiomatization is possible even for finite monoids.

ABSTRACT

It is shown that a finite monoid can have an infinite irredundant basis of equations.

Motivation & Objective

• To resolve Problem 1.1, which asks whether a finite monoid can have an infinite irredundant basis of monoid identities.
• To demonstrate that infinite irredundant equational axiomatization is possible in the monoid signature, despite the strong deductive power of the identity element.
• To construct a specific finite nilpotent monoid whose equational theory is not finitely based and has no redundant equations in any complete axiomatization.
• To extend prior results on infinite irredundant bases in semigroups to the more restrictive monoid setting, where such properties are expected to be rarer.

Proposed method

• The construction relies on analyzing word structures in equations, particularly focusing on variable occurrences, subwords, and the role of the identity element in simplifying long equations.
• A substitution mapping φ is inductively defined to reduce complex words to simpler forms, preserving equational consequences while tracking variable positions and occurrences.
• The proof uses a recursive inductive construction of substitutions that maintain structural invariants, such as the relative positions of first and second occurrences of variables.
• Key claims (e.g., Claim 3, Claim 8) are used to constrain possible letter sequences in words, ensuring that certain subword patterns must occur under given conditions.
• The method shows that long equations in the basis cannot be derived from shorter ones, and no single equation can be removed without losing completeness.
• The argument establishes that the equational theory of the constructed monoid is not finitely based and that every complete axiomatization must be infinite and irredundant.

Experimental results

Research questions

• RQ1 Can a finite monoid have an infinite irredundant basis of monoid identities, despite the strong deductive influence of the identity element?
• RQ2 Is it possible for a finite algebra to have an equational theory that is not finitely based and yet admits no redundant equations in any complete axiomatization?
• RQ3 How does the presence of the identity element in monoids affect the possibility of infinite irredundant axiomatization compared to semigroups?
• RQ4 What structural properties of words and variable occurrences in equations prevent short equations from deriving long ones in finite monoids?
• RQ5 Does the existence of an infinite irredundant equational basis in a finite monoid imply that its variety has continuum many subvarieties?

Key findings

• A finite nilpotent monoid is explicitly constructed whose equational theory has no finite axiomatization and admits no redundant equations in any complete basis.
• The equational theory of the constructed monoid is shown to be infinite and irredundant, meaning every equation in any complete basis is logically necessary and cannot be derived from others.
• The proof establishes that long equations in the basis cannot be derived from shorter ones, due to the specific structure of variable occurrences and subword patterns.
• The construction demonstrates that the presence of the identity element does not necessarily destroy the possibility of infinite irredundant axiomatization, contrary to expectations.
• The example is contained in many finitely based varieties, yet its own equational theory remains infinitely and irredundantly axiomatizable.
• The result confirms that infinite irredundant equational axiomatization is possible in the monoid signature, solving Problem 1.1 affirmatively.