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Automata theory has played an important role in modeling behavior of systems since last couple of decades. The algebraic automaton has emerged with several modern applications because of having properties and structures from algebraic theory. Design of a complex system not only requires behavior but it also needs to model its functionality. Z notation is an ideal one used for describing functionality. Consequently, an integration of algebraic automata and Z will be an effective tool for modeling of complex systems. In this paper, we have combined algebraic automata and Z defining a relationship between fundamentals of these approaches. At first, we have described extended form of algebraic automaton. Then the concepts of homomorphism and its variants are defined over strongly connected automata. Finally, monoid endomorphisms and group automorphisms are defined, and formal proof of their equivalence is given under certain assumptions. The specification is analyzed and validated using Z/EVES tool.

Automata theory has played an important role in theoretical computer science since last couple of decades. The alge-braic automaton has emerged with several modern applications, for example, optimization of programs, design of model checkers, development of theorem provers because of having certain interesting properties and structures from algebraic theory of mathematics. Design of a complex system requires functionality and also needs to model its control behavior. Z notation has proved to be an effective tool for describing state space of a system and then defining operations over it. Consequently, an integration of algebraic automata and Z will be a useful computer tool which can be used for modeling of complex systems. In this paper, we have linked algebraic automata and Z defining a relationship between fundamentals of these approaches which is refinement of our previous work. At first, we have described strongly connected algebraic automata. Then homomorphism and its variants over strongly connected automata are specified. Next, monoid endomorphisms and group automorphisms are formalized. Finally, equivalence of endomorphisms and automorphisms under certain assumptions are described. The specification is analyzed and validated using Z/Eves toolset

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