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Decompositions of automata and transition semigroups
Tatjana Petkovi\'c3, Miroslav \'Ciri\'c4, Stojan Bogdanovi\'c5
Abstract
The purpose of this paper is to describe structural properties of automata whose transition semigroups have a zero, left zero, right zero or bi-zero, or are nilpotent extensions of rectangular bands, left zero bands or right zero bands, or are nilpotent. To describe the structure of these automata we use various well-known decomposition methods of automata theory - direct sum decompositions, subdirect and parallel decompositions, and extensions of automata. Automata that appear as the components in these decompositions belong to some well-known classes of automata, such as directable, definite, reverse definite, generalized definite and nilpotent automata. But, we also introduce some new classes of automata: generalized directable, trapped, one-trapped, locally directable, locally one-trapped, locally nilpotent and locally definite automata. We explain relationships between the classes of all these automata.
Keywords: automaton, transition semigroup, direct sum decomposition, directable automata, trapped automata, generalized directable automata, locally directable automata, generalized varieties.
Footnotes:
1 Supported by Grant 04M03B of RFNS through Math. Inst. SANU.
2 Supported by Grant 04M03B of RFNS through Math. Inst. SANU.
3 University of Ni s, Faculty of Philosophy, \' Cirila i Metodija 2, P. O. Box 91, 18000 Ni s, Yugoslavia, e-mail: tanjapetarchimed.filfak.ni.ac.yu
4 University of Ni s, Faculty of Philosophy, \' Cirila i Metodija 2, P. O. Box 91, 18000 Ni s, Yugoslavia, e-mail: mciricarchimed.filfak.ni.ac.yu
5 University of Ni s, Faculty of Economics, Trg VJ 11, P. O. Box 121, 18000 Ni s, Yugoslavia, e-mail: sbogdanarchimed.filfak.ni.ac.yu