Free lattices, communication and money games

Is cost a kind of truth value? What kind of communication is trading? One goal of this work is to show that the theory of free lattices can shed some light on these questions. Another goal is to lay the foundation of a theory of communication based on free lattices and, in further work, on free bicompletions of categories. Here we discuss the game theoretic interpretation of the theory of free lattices. To each term A representing an element of a free lattice we associate a game G(A) with two players, sigma(A) and pi(A) (symbolically sigma(A):G(A):pi(A)) To any pair (A, B) of such terms we associate a game (G(A),G(B)) having two components or boards, G(A) and G(B), and three players, a mediator sigma facing two opposites, sigma(A) and pi(B), one on each board (symbolically sigma(A):G(A):sigma:G(B):pi(B)). A strategy for the mediator is viewed as a communication strategy between the opposites. A communication strategy is complete if it is a winning strategy. We show that the mediator has a complete communication strategy iff A less than or equal to B in the free lattice. The transitivity of the relation A I B is then a consequence of the fact that communication strategies compose. We formally extend these results to lattices enriched over a quantale viewed as a domain of generalized logical values. In the special case of the quantale [0, infinity] of (extended) non-negative real numbers we obtain a theory of free metric lattices. The terms representing the elements of a free metric lattice are two person games in which the players each have a stake from which they exchange numerical values, called money. For any pair of such terms we construct a money game (G(A),G(B)) with three players, a mediator trading with two opposites or sides. The cost of the least expensive winning strategy a mediator can have on (G(A),G(B)) is a distance d(A, B) defining the metric of the free metric lattice. In the last section we describe certain operations on games that are related to the connectives of linear logic.(1)