[Paper Review] Groebner-Shirshov bases for free inverse semigroups

L. A. Bokut, Yuqun Chen+1 more|arXiv (Cornell University)|Apr 7, 2008

semigroups and automata theory29 references3 citations

TL;DR

This paper establishes Gröbner-Shirshov bases for free inverse semigroups using the deg-lex monomial ordering, building on Poliakova and Schein's 2005 construction of canonical idempotents. It provides a unique, shortest normal form for each element and a constructive algorithm to reduce any word to this normal form, ensuring uniqueness and efficiency in word reduction within free inverse semigroups.

ABSTRACT

A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.

Motivation & Objective

• To develop a constructive normal form for elements in free inverse semigroups using Gröbner-Shirshov bases.
• To resolve the problem of non-uniqueness in word representations by identifying a unique, shortest normal form.
• To provide an algorithmic method to transform any word into its unique normal form using the Gröbner-Shirshov basis.
• To extend the applicability of Gröbner-Shirshov theory to inverse semigroups by leveraging canonical idempotents from Poliakova and Schein's framework.

Proposed method

• Utilize the deg-lex monomial ordering on the free monoid generated by X ∪ X⁻¹ to define leading words and compositions.
• Construct a Gröbner-Shirshov basis S from defining relations involving canonical idempotents derived from Poliakova and Schein’s construction.
• Apply the Composition-Diamond Lemma to prove that all compositions (inclusion and intersection) are trivial modulo S, ensuring S is a Gröbner-Shirshov basis.
• Define the set of irreducible words Irr(S) as the normal forms, characterized by absence of subwords of the form aā⁻¹ and proper ordering of canonical idempotents.
• Ensure uniqueness by requiring that the first and last letters of idempotent factors do not match adjacent word parts.
• Implement the Gröbner-Shirshov algorithm to reduce any word to its unique normal form via successive reductions using the basis S.

Experimental results

Research questions

• RQ1 Can a unique and shortest normal form be systematically derived for elements in a free inverse semigroup using Gröbner-Shirshov bases?
• RQ2 How can the deg-lex ordering be applied to construct a Gröbner-Shirshov basis for free inverse semigroups?
• RQ3 What role do canonical idempotents play in ensuring the uniqueness of normal forms in inverse semigroups?
• RQ4 Are all compositions (inclusion and intersection) trivial modulo the constructed basis, confirming it as a Gröbner-Shirshov basis?
• RQ5 Can the Gröbner-Shirshov algorithm be effectively used to reduce any word to its unique normal form in free inverse semigroups?

Key findings

• The set of irreducible words Irr(S) forms a unique and shortest normal form for each element in the free inverse semigroup FI(X).
• The normal forms are characterized by the absence of subwords of the form yy⁻¹ for y ∈ X ∪ X⁻¹ and proper ordering of canonical idempotents.
• All inclusion and intersection compositions in the basis S are trivial modulo S, confirming that S is a Gröbner-Shirshov basis.
• The Gröbner-Shirshov algorithm provides a systematic and effective method to reduce any word to its unique normal form.
• The normal forms are a subset of the canonical words defined by Poliakova and Schein, but are uniquely determined per element, unlike general canonical words.
• The construction ensures that different words in Irr(S) represent distinct elements in FI(X), guaranteeing injectivity of the normal form map.