Abstract
In this paper, we consider the multiplicative groupoid of matrices with elements in a lattice with 0 and 1. Examples of such groupoids are the semigroup of binary relations and semigroups of minimax (fuzzy) relations. It is shown that every automorphism of a groupoid is the composition of an inner automorphism and the automorphism defined by an automorphism of the lattice. Despite the fact that, in general, the groupoid is not associative, it satisfies the UA-property: Every multiplicative automorphism is an additive automorphism. Earlier, the realization of the UA-property has been considered mainly for associative rings and semirings. We describe the invertible matrices that define inner automorphisms.
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References
I. I. Artamonova, “On the uniqueness of addition in semirings,” Fundam. Prikl. Mat., 3, No. 4, 1093–1100 (1997).
G. Birkhoff, Theory of Lattices [Russian translation], Nauka, Moscow (1984).
L. M. Gluskin, “Automorphisms of semigroups of binary relations,” Mat. Zapiski Ural. Gos. Univ., 6, 44–54 (1967).
G. Gr¨atzer, General Lattice Theory [Russian translation], Mir, Moscow (1982).
J. Giveon, “Lattice matrices,” Inform. Control, 7, 477–484 (1964).
R. E. Jonsson, “Ring with unique addition,” Proc. Am. Math. Soc., 9, 55–61 (1958).
A. Kofman, Introduction into the Theory of Fuzzy Sets [in Russian], Radio i Svyaz’, Moscow (1982).
V. G. Kumarov and E. E. Marenich, “Inversion of matrices over lattices with pseudocompliments,” Fundam. Prikl. Mat., 11, No. 3, 139–154 (2005).
A. V. Mikhalev, “Multiplicative classification of associative rings,” Mat. Sb., 135, No. 2, 210–233 (1988).
S. E. Ricart, “One-to-one mappings of rings and lattices,” Bull. Am. Math. Soc., 54, 578–764 (1948).
V. D. Shmatkov, “Algebras of incidence over lattices,” Usp. Mat. Nauk, 47, No. 4, 217–218 (1992).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 19, No. 1, pp. 195–204, 2014.
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Shmatkov, V.D. Isomorphisms and Automorphisms of Matrix Algebras Over Lattices. J Math Sci 211, 434–440 (2015). https://doi.org/10.1007/s10958-015-2614-z
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DOI: https://doi.org/10.1007/s10958-015-2614-z