Dense Relations Are Determined by Their Endomorphism Monoids Jo˜aoAra´ ujo and Janusz Konieczny Abstract We introduce the class of dense relations on a set X and prove that for any finitary or infinitary dense relation ρ on X, the relational system (X, ρ) is determined up to semi-isomorphism by the monoid End (X, ρ) of endomorphisms of (X, ρ). In the case of binary relations, a semi-isomorphism is an isomorphism or an anti-isomorphism. 2000 Mathematics Subject Classification : 20M20, 20M15. 1 Introduction For a mathematical structure M , let End(M ) denote the endomorphism monoid of M . A general problem, which attracted a considerable attention, can be stated as follows: Let M 1 ,M 2 be two mathematical structures. Given that End(M 1 ) ª = End(M 2 ), what can we say about the relation between the structures M 1 and M 2 themselves? For example, Schein [8] proved that if M 1 and M 2 are two partially ordered sets, semilattices, distributive lattices, or Boolean algebras, then End(M 1 ) ª = End(M 2 ) if and only if M 1 and M 2 are isomorphic or anti-isomorphic. For other results of this kind, see [2], [4], and [5]. The aim of this note is to prove a similar result for the class of dense relations. These relations include partial orders, binary relations that are reflexive and symmetric, and generalized equivalence relations. Let I be an arbitrary non-empty index set. An I -tuple of elements of a set X is a mapping f : I ! X . If I is finite with |I | = n, we shall assume that I = {1, 2,...,n}, denote f : I ! X by (1f, 2f,...,nf ), and refer to f as an n-tuple. An I -relation ρ on X is any set of I -tuples of elements of X . If |I | = n, an I -relation ρ is an n-ary relation on X , that is, a set of n-tuples of elements of X . By a relation on X , we shall mean an I -relation on X for some index set I . Let ρ be an I -relation on X . An endomorphism of a relational system (X, ρ) is a mapping a : X ! X that preserves ρ, that is, fa 2 ρ for every f 2 ρ, where fa : I ! X is the composition of f : I ! X and a : X ! X . (We compose from left to right, that is, i(fa)=(if )a for i 2 I .) If ρ is an n-ary relation, we can use the n-tuple notation. With that notation, we have that a : X ! X is an endomorphism of (X, ρ) if and only if (x 1 a,...,x n a) 2 ρ for every (x 1 ,...,x n ) 2 ρ. We denote by End (X, ρ) the monoid of endomorphisms of (X, ρ). Turning to the definition of a dense relation, we denote by ρ § the set of all mappings f : I ! X such that σf/ 2 ρ for every σ 2 S (I ), where S (I ) is the symmetric group of I . 1