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Linear lambda terms as invariants of rooted trivalent maps. (English) Zbl 1420.68050
Summary: The main aim of the paper is to give a simple and conceptual account for the correspondence (originally described by O. Bodini et al. [Theor. Comput. Sci. 502, 227–238 (2013; Zbl 1297.68184)]) between $$\alpha$$-equivalence classes of closed linear lambda terms and isomorphism classes of rooted trivalent maps on compact-oriented surfaces without boundary, as an instance of a more general correspondence between linear lambda terms with a context of free variables and rooted trivalent maps with a boundary of free edges. We begin by recalling a familiar diagrammatic representation for linear lambda terms, while at the same time explaining how such diagrams may be read formally as a notation for endomorphisms of a reflexive object in a symmetric monoidal closed (bi)category. From there, the “easy” direction of the correspondence is a simple forgetful operation which erases annotations on the diagram of a linear lambda term to produce a rooted trivalent map. The other direction views linear lambda terms as complete invariants of their underlying rooted trivalent maps, reconstructing the missing information through a Tutte-style topological recurrence on maps with free edges. As an application in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent maps as linear lambda terms containing no closed proper subterms, and conclude by giving a natural reformulation of the Four Color Theorem as a statement about typing in lambda calculus.

##### MSC:
 68N18 Functional programming and lambda calculus 03B40 Combinatory logic and lambda calculus 03G30 Categorical logic, topoi 05A16 Asymptotic enumeration 05C15 Coloring of graphs and hypergraphs
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