This paper investigates the algebraic structure of Krom logic programs, consisting only of facts and rules with at most one body atom. We show that sequential composition endows the class of Krom programs with a natural monoid structure and that this structure admits rich algebraic extensions to Krom seminearrings, Krom quemirings, Krom-Conway seminearrings, and Krom-Conway omegaseminearrings. Furthermore, we establish explicit generating sets and canonical decompositions, study the associated ${}^ω$-operator, characterize the Kleene star in graph-theoretic terms, and relate finite Krom monoids to transformation monoids and finite-state automata. These results provide new connections between logic programming, algebraic automata theory, and algebraic graph theory.

翻译：本文研究由事实和最多包含一个体原子规则组成的Krom逻辑程序的代数结构。我们证明，序贯复合为Krom程序类赋予了自然的幺半群结构，且该结构能够扩展出丰富的代数形式，包括Krom半近环、Krom拟环、Krom-康威半近环以及Krom-康威欧米伽半近环。此外，我们建立了显式生成集与标准分解，研究了关联的${}^\omega$算子，从图论角度刻画了Kleene星，并揭示了有限Krom幺半群与变换幺半群及有限状态自动机之间的联系。这些结果为逻辑程序设计、代数自动机理论和代数图论之间建立了新的关联。