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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