Total relation

In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y  x R y). Serial relations are sometimes called total relations. In set theory, a serial relation is a binary relation R for which every element of the domain has a corresponding range element (∀ x ∃ y  x R y). Serial relations are sometimes called total relations. For example, in ℕ = natural numbers, the 'less than' relation (<) is serial. On its domain, a function is serial. A reflexive relation is a serial relation but the converse is not true. However, a serial relation that is symmetric and transitive can be shown to be reflexive. In this case the relation is an equivalence relation. If a strict order is serial, then it has no maximal element. In Euclidean and affine geometry, the serial property of the relation of parallel lines ( m ∥ n ) {displaystyle (mparallel n)} is expressed by Playfair's axiom. In Principia Mathematica, Bertrand Russell and A. N. Whitehead refer to 'relations which generate a series' as serial relations. Their notion differs from this article in that the relation may have a finite range. For a relation R let {y: xRy } denote the 'successor neighborhood' of x. A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood. Similarly an inverse serial relation is a relation in which every element has non-empty 'predecessor neighborhood'. In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D. Serial relations can be characterized algebraically by equalities and inequalities about relation compositions. If R ⊆ X × Y {displaystyle Rsubseteq X imes Y} and S ⊆ Y × Z {displaystyle Ssubseteq Y imes Z} are two binary relations, then their composition S ∘ R {displaystyle Scirc R} is defined as the relation S ∘ R = { ( x , z ) ∈ X × Z ∣ ∃ y ∈ Y : ( x , y ) ∈ R ∧ ( y , z ) ∈ S } . {displaystyle Scirc R={(x,z)in X imes Zmid exists yin Y:(x,y)in Rland (y,z)in S}.}