On Epimorphisms of Hypergraphic Automata and Input Symbol Semigroups

Abstract

Hypergraphic automata are automata whose state sets and output symbol sets are hypergraphs invariant under the actions of the transition and output functions. Universally attracting objects in the category of such automata are called universal hypergraphic automata; their semigroups of input symbols are algebras of mappings whose properties are tightly linked to the algebraic structure of the automata themselves. In this paper we describe surjective morphisms (epimorphisms) of universal hypergraphic automata and of their semigroups of input symbols, for effective hypergraphs with $p$-definable edges. Under two mild hypotheses, a triple $(f,{\mathbb{P}}_s,g)$ is an epimorphism if and only if the state component $f$ is an isomorphism and the output component $g$ is a homomorphism satisfying a lifting condition. We introduce weak and strong epimorphisms of hypergraphs, show that they differ, and prove that liftability yields strong surjectivity under a size condition on edges. For automata whose state hypergraphs are projective or affine planes, the state components of epimorphisms are exactly the collineations.

arXiv comment13 pages