Incidence rings of pre-ordered sets.
W. Russell Belding · Notre Dame Journal of Formal Logic · 1973
Introduction.In this paper* every relation Ion a set X is a binary relation which is transitive and reflexive.G. C. Rota [2] has defined incidence rings of partially ordered systems (X, ύ)..We generalize these rings by dropping the anti-symmetric condition on the order % If X is a set and••% a binary relation on X, then (X, %) shall denote this relational system.We say that (X Γ S) is a pre-ordered relational system if the relation = is transitive and reflexive.; -If confusion is unlikely, then we shall often take the liberty of using the relation ^ to denote the usual ordering of the natural numbers and also to denote a relation on a set X. Unless otherwise stated 0,1 should be understood to be real numbers.To each relational system (X, =) there is a unique zeta function, ζ^ mapping X xX into {0,1}, For ,x, ye X, ζ(x,y) = 1,.. if--AT ^ y znd ζ(x,y) = 0 otherwise.In the context of a relation system (X> =),[x,y] =-{ue x\x = u ~ y} is an interval and (X, ^) is locally finite'iff every such interval is empty or a finite set.We shall consider only rings R which have a multiplicative identity; rings may or may not be commutative.We do not assume any relationship between the rings R and sets X we discuss.The symbol β* denotes the set of units of the ring R; the function det is the determinant function, lί n is a positive integer,, then bΛ{n,R\ denotes the complete ring of n x n matrices over the ring R. If X is any set, then £χ denotes the group of permutations of the set X\ for positive integers n, ^denotes S[ 1)ttt>n \.For a given ring R and locally finite pre-ordered system {X, ύ), the incidence ring 1= (X, =, R) is set-theoretically the set of functions / mapping X x X into R satisfying the following order condition.For every x, ye X,f(x,y) Φ 0 only if x ύ y.Multiplication, addition and scalar multiplication for incidence rings are defined in section 1.If [x, y\ is a