Distance in lattices
by
Jobst Heitzig
Institut für Mathematik, Universität Hannover, Welfengarten 1,30167 Hannover,Germany.

As on other algebraic objects there are natural ``internal'' distances on different kinds of lattices, like the distance |x-y| in abelian l-groups. Other examples arise in logics: the implication x --> y in a Heyting algebra is a non-symmetric and non-strict distance, while the bi-conditional x <--> y=(x \/ y^c) /\ (x^c \/ y) in an orthomodular lattice is symmetric. A common generalization of both is the biconditional in Gevers' and Reinhold's hypo-implication algebras.

I will analyse what the homometries (=maps that preserve inequalities between sums of distances) are in these examples, and discuss the corresponding functors into the category of all distance sets (X, d, M) (where M is a quasi-ordered monoid, d:X×X --> M, d(x, x)=0, and d(x, y)+d(y, z) >= d(x, z)) with homometries.

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Date received: May 12, 2000