×

zbMATH — the first resource for mathematics

Lexicographic product decompositions of partially ordered quasigroups. (English) Zbl 0986.06012
Let \(Q\) be a partially ordered quasigroup with an idempotent element \(h\). The author defines the notion of the lexicographic product decomposition of \(Q\) with respect to the element \(h\). The main result of the paper says that if \(Q=(A_1\circ A_2\circ \dots \circ A_n)_h\) and \(Q=(B_1\circ B_2\circ \dots \circ B_m)_h\) are such lexicographic product decompositions and if all factors \(A_i, B_j\) \((i=1,\dots , n; j=1,\dots ,m)\) are directed, then the given lexicographic product decompositions of \(Q\) have isomorphic refinements.

MSC:
06F15 Ordered groups
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] BELYAVSKAYA G. B.: Direct decomposition of quasigroups. Mat. Issled. 5 (1987), 23-38. · Zbl 0635.20036
[2] BELYAVSKAYA G. B.: Complete direct decompositions of quasigroups with an idempotent. Mat. Issled. 113 (1990), 21-36. · Zbl 0746.20053
[3] BELOUSOV V. D.: Foundations of the Theory of Quasigroups and Loops. Nauka, Moscow, 1967.
[4] BIRKHOFF G.: Lattice Theory. Amer. Math. Soc, Providence, RI, 1967. · Zbl 0153.02501
[5] ČERNÁK Š.: Lexicographic products of cyclically ordered groups. Math. Slovaca 45 (1995), 29-38. · Zbl 0836.06010
[6] JAKUBIK J.: Lexicographic products of partially ordered groupoids. Czechoslovak Math. J. 14(89) (1964), 281-305.
[7] JAKUBÍK J.: Lexicographic product decompositions of cyclically ordered groups. Czechoslovak Math. J. 48(123) (1998), 229-241. · Zbl 0952.06021
[8] JAKUBÍK J.: Lexicographic products of half linearly ordered groups. Czechoslovak Math. J. · Zbl 1079.06504
[9] MAL’CEV A. I.: On ordered groups. Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 473-482.
[10] TARARIN V. M.: Ordered quasigroups. Izv. Vyssh. Uchebn. Zaved. Mat. 1 (1979), 82-86.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.