Combinatorial aspects of relations. (English) Zbl 0549.20059

A multivalued group-like system called polygroup (a special type of multigroup in the sense of M. Dresher and Ø. Ore [Am. J. Math. 60, 705-733 (1938; Zbl 0019.10701)] is introduced. Polygroups can be characterized as the atomic structures (under composition and reverse) of complete atomic integral relation algebras (see B. Jónsson and A. Tarski [ibid. 74, 127-162 (1952; Zbl 0049.15801)]. Due to this, there is a correspondence between certain properties of IRA’s (such as different kinds of representability) and those of polygroups. Also, information can be obtained about automorphism groups of colour schemes (colourings of the edges of complete graphs, subject to some conditions) from the knowledge of polygroups attached to them (so called chromatic polygroups). The last two theorems give some approach to problems concerning the orbits of stabilizers of the elements in transitive permutation groups by ways of polygroups.
Reviewer: G.Pollák


20N99 Other generalizations of groups
05C15 Coloring of graphs and hypergraphs
08A02 Relational systems, laws of composition
20B20 Multiply transitive finite groups
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[1] N. L.Biggs,Algebraic Graph Theory, Cambridge Univ. Press, 1974. · Zbl 0284.05101
[2] R. C. Bose andD. M. Mesner,On linear associative algebras corresponding to association schemes of partially balanced designs. Annals Math. Stat.36 (1959), 21-38. · Zbl 0089.15002
[3] P. J.Cameron and J. H.Van Lint,Graph Theory, Coding Theory and Block Designs, London Math. Soc. Lecture Notes no. 19, Cambridge Univ. Press, 1975.
[4] S. D. Comer,The representation of a class of simple CA3’s. Notices AMS23 (1976), A-41.
[5] S. D. Comer,Integral relation algebras via pseudogroups. Notices AMS23 (1976), A-659.
[6] S. D. Comer,Quasigroup representative CA 3’sare not finitely axiomatizable. Notices AMS24 (1977), A-43.
[7] S. D.Comer,Multivalued loops and their connection with algebraic logic. Manuscript, 173 pages, 1979.
[8] M. Dresher andO. Ore,Theory of multigroups. Amer. J. Math.60 (1938), 705-733. · JFM 64.0056.01
[9] D. G. Higman,Combinatorial considerations about permutation groups. Lecture Notes. Mathematical Institute, Oxford, 1972.
[10] B. Jónsson andA. Tarski,Boolean Algebras with Operators II. Amer. J. Math.74 (1952), 127-162. · Zbl 0049.15801
[11] B. Jónsson,Representation of modular lattices and of relation algebras. Trans. AMS92 (1959), 449-464. · Zbl 0105.25302
[12] R. C. Lyndon,Relation algebras and projective geometries. Michigan Math. J.8 (1961), 21-28. · Zbl 0105.25303
[13] R.McKenzie,The representation of relation algebras. Thesis. University of Colorado, 1966.
[14] R. McKenzie,Representations of integral relation algebras. Michigan Math. J.17 (1970), 279-287. · Zbl 0197.29202
[15] J. D. Monk,Model-theoretic methods and results in the theory of cylindric algebras. In, ?Theory of Models,? Proceed. 1963 Internat. Symp. at Berkeley, North-Holland Pub. Co., 1965, 238-250.
[16] J. D. Monk,On representable relation algebras. Michigan Math. J.11 (1964), 207-210. · Zbl 0137.00603
[17] P. M.Neumann,Finite permutation groups, edge-colored graphs and matrices. In, ?Topics in Group Theory and Computation? (Editor, Michael P. J. Curran), Academic Press, 1977.
[18] W. Prenowitz,Projective geometries as multigroups. Amer. J. Math.65 (1943), 235-256. · Zbl 0063.06336
[19] W.Prenowitz and J.Jantosciak,Join Geometry. Springer-Verlag, 1979.
[20] C. C. Sims,Graphs and finite permutation groups. Math. Zeitschrift95 (1967), 76-86. · Zbl 0244.20001
[21] Y. Utumi,On hypergroups of group right cosets, Osaka Math. J.1 (1949), 73-80. · Zbl 0036.29302
[22] M. J. Weiss,On simply transitive groups, Bull. AMS40 (1935), 401-405. · Zbl 0009.30003
[23] H.Wielandt,Permutation groups through invariant relations and invariant functions. Lecture Notes. Ohio State University, 1969. · Zbl 0165.03901
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