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The lattice of idempotent binary relations

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Abstract

Order-theoretic properties of the complete latticeE(A) of indempotent binary relations ρ=ρ2 on the given setA are investigated. The elements ρ ofE(A) are classified according to theirfixed field I(ρ)={a∈A|(a, a)∈ρ} as being either offinite type, dense, or ofmixed type. When |A|>1E(A) is a non-atomic, non-coatomic lattice in which each element is a meet of meet-irreducible elements. The elements ofE(A) which are joins of join-irreducible elements form a compactly generated complete latticeF(A) which is a join-sublattice ofE(A) consisting of all elements having finite type. The setsD(A), M(A) of elements ρ ofE(A) which are dense (i.e., satisfy ρ≠ϕ andI(ρ)=ϕ) or of mixed type (i.e., are neither dense nor of finite type) resp. are non-empty only when |A| is infinite.D(A) is a partial meet-subsemilattice ofE(A) admitting no minimal elements. The group of order automorphisms of the latticeE(A) is isomorphic toS A ×Z 2 and each order automorphism ofE(A) preserves inverses.

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  1. Technion, Israel Institute of Technology, Technion City, Haifa, Israel

    Zahava Shmuely

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  1. Zahava Shmuely
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Shmuely, Z. The lattice of idempotent binary relations. Algebra Universalis 9, 297–303 (1979). https://doi.org/10.1007/BF02488041

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  • DOI: https://doi.org/10.1007/BF02488041

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