On some affine semigroups characterized by a finite-state automata

TL;DR

This work defines $P$-semigroups inside the framework of $\\mathcal{C}$-semigroups and fixes a monomial order to study invariants such as genus, Frobenius element, and multiplicity. It introduces a finite-state automaton $M(S)$ that recognizes paths along $P$ inside $S$, and proves that $S$ is a $P$-semigroup iff every recognized word can be extended by an element of $P$, establishing a precise automata-theoretic criterion. The authors develop tree-based, algorithmic frameworks to enumerate all $P$-semigroups with prescribed genus, Frobenius element, or multiplicity, including detailed descriptions of the corresponding graphs and practical generation procedures, with illustrative examples. By linking semigroup properties to automata and providing concrete enumeration algorithms, the paper bridges affine semigroup theory and discrete mathematics, enabling computational classification of $P$-semigroups under various invariants.

Abstract

This work introduces a new kind of affine semigroups called $P$-semigroups. Within the framework of $\mathcal C$-semigroups, we define a finite-state automaton associated to them. Moreover, this automaton determines whether a $\mathcal C$-semigroup is a $P$-semigroup, which represents a bridge between affine semigroups and Discrete Mathematics. Furthermore, some algorithms for computing all the $P$-semigroups with a fixed Frobenius element, genus, or multiplicity are provided.

Figures (6)

• Figure 1: A $P$-semigroup
• Figure 2: A path inside the $P$-semigroup
• Figure 3: The first three levels of $G\left(\mathcal{S}(P)\right)$
• Figure 4: The tree of $G\left(\mathcal{S}(P,f)\right)$
• Figure 5: The set $\mathcal{S}(P)_3$

Theorems & Definitions (28)

• Proposition 1
• proof
• Example 2
• Definition 3
• Definition 4
• Remark 5
• Example 6
• Lemma 7
• Theorem 8
• proof