Abstract
Let\(\mathfrak{X}\)=〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC(\(\mathfrak{X}\)) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC(\(\mathfrak{X}\))U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.
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References
T. Evans, et al.Logic Colloquium 73 (Bristol, 1973).Studies in Logic and the foundations of Mathematics, Vol. 80, North-Holland, Amsterdam, 1975.
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Płonka, J. On lattices of congruences of relational systems and universal algebras. Algebra Universalis 13, 82–88 (1981). https://doi.org/10.1007/BF02483825
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DOI: https://doi.org/10.1007/BF02483825