Mathematics for linguists

Gerhard J¨ager

gerhard.jaeger@uni-tuebingen.de

Uni T¨ubingen, WS 2009/2010

November 5, 2009

Weak orderings

A relationR is a weak ordering iff R is

• transitive,

• reflexive, and

• anti-symmetric

Strict ordering

A relationR is a strict orderings iff R is

• transitive,

• irreflexive, and

Examples:

• A={a, b, c, d}

• R₁={ha, bi,ha, ci,ha, di,hb, ci,ha, ai,hb, bi,hc, ci,hd, di}

• R₂={hb, ai,hb, bi,ha, ai,hc, ci,hd, di,hc, bi,hc, ai}

• R3=

{hd, ci,hd, bi,hd, ai,hc, bi,hc, ai,ha, ai,hb, bi,hc, ci,hd, di,hb, ai}

corresponding strict orderings:

• S1 ={ha, bi,ha, ci,ha, di,hb, ci}

• S2 ={hb, ai,hc, bi,hc, ai}

• S₃ ={hd, ci,hd, bi,hd, ai,hc, bi,hc, ai,hb, ai}

A weak orderingR⊆A×A and a strict ordering S correspond to each other iff

R=S∪id_A

• further examples:

• ≤and<⊆ N×N

• ⊆and⊂ are subsets of ℘(A)×℘(A)

Terminology

LetR be an ordering (strict or weak).

• ais a predecessor ofb iff R(a, b).

• ais a successor of biff R(b, a).

• ais a immediate predecessor ofb iff

• a6=b,

• R(a, b), and

• there is noc6∈ {a, b} such thatR(a, c)andR(c, b).

• ais an immediate successor ofb iff bis an immediate predecessor ofa.

LetR⊆A×Abe an ordering (strict or weak).

• An element x∈A isminimal iff there is noy 6=x which is a predecessor ofx.

• An element x∈A isleast iff x is the predecessor of all other elements of A.

• An element x∈A ismaximal iff there is noy 6=x which is a successor of x.

• An element x∈A isgreatest iff x is the sucessor of all other elements of A.

An ordering has at most one least and at most one greatest element, but ther may be any number of minimal or maximal

Linear ordering

An ordering islinear(or total) iff it is connected.

If an orderingR is not linear, there are two elementsa andbsuch that neitherR(a, b)norR(b, a). The ordering is thus not complete, but partial.

• further kinds of orderings:

• An orderingR⊆A×Aiswell-foundediff every restriction R∩(B×B)to a subsetB⊂Acontains a minimal element.

To put it another way: a well-founded ordering does not have infinitely descending branches.

• An orderingR⊆A×Aisdense iff for any pair of elementsx andywithR(x, y), there is a third elementz, which differs fromxandy, such thatR(x, z)andR(z, y).