Globalization of partial monoid actions via abstract rewriting systems

2025/12/24 by Mykola Khrypchenko, Khrypchenko, Mykola, Francisco Klock +1
Computer Science · Mathematics · #16S10 #16S15 #16W22 #20M05 #20M30 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory

paper · doi:10.48550/arxiv.2512.21173

openalex publication_date 2025/12/24 · openalex created_date 2025/12/26 · openalex updated_date 2026/07/28

Abstract

We study the globalization problem for a strong partial action α of a monoid M on a semigroup X via the associated rewriting system (X_M⁺,→). We show that the local confluence of (X_M⁺,→) is sufficient for the globalizability of α but, unlike the group case, it is not necessary. Focusing on the monoid M=G⁰, where G is a group, we obtain an explicit criterion for the globalizability of α and a criterion for the local confluence of (X_M⁺,→). Several applications to strong partial actions of the monoid M=\0,1\ on semigroups and algebras, as well as to strong partial actions of an arbitrary monoid M on left zero and null semigroups, are presented.

Citations

• On the homology of partial group representations
• Partial Groupoid Actions on Smooth Manifolds
• Globalization of partial inverse semigroupoid actions on sets
• Partial monoid actions on objects in categories with pullbacks and their globalizations
• Restriction and Extension of Partial Actions
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• Geometrically Partial actions
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Related

• Partial Semigroupoid Actions on Sets
• Reflectors and globalizations of partial actions of groups
• Globalization of partial inverse semigroupoid actions on sets
• Partial monoid actions on objects in categories with pullbacks and their globalizations
• Partial Category Actions on Sets and Topological Spaces