Abstract A perfect code in a graph $$\Gamma = (V, E)$$ Γ = ( V , E ) is a subset C of V such that no two vertices in C are adjacent and every vertex in $$V \setminus C$$ V \ C is adjacent to exactly one vertex in C . A total perfect code in $$\Gamma $$ Γ is a subset C of V such that every vertex of $$\Gamma $$ Γ is adjacent to exactly one vertex in C . In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

Abstract In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have $$m+r$$ m + r partitions with $$r\ge 2$$ r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If $$m=2$$ m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $$m>2$$ m > 2 , things are more restricted. Any $$m+1$$ m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality $$m+r$$ m + r in the $$(m-1)$$ ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.

We can use the compositional semantics of Hodges [9] to show that any compositional semantics for logics of imperfect information must obey certain constraints on the number of semantically inequivalent formulas. As a corollary, there is no compositional semantics for the ‘independence-friendly’ logic of Hintikka and Sandu (henceforth IF) in which the interpretation in a structure A of each 1 -ary formula is a subset of the domain of A (Corollary 6.2 below proves this and more). After a fashion, this rescues a claim of Hintikka and provides the proof which he lacked: … there is no realistic hope of formulating compositional truth-conditions for [sentences of IF], even though I have not given a strict impossibility proof to that effect. (Hintikka [6] page 110ff.) One curious spinoff is that there is a structure of cardinality 6 on which the logic of Hintikka and Sandu gives nearly eight million inequivalent formulas in one free variable (which is more than the population of Finland). We thank the referee for a sensible change of notation, and Joel Berman and Stan Burris for bringing us up to date with the computation of Dedekind's function (see section 4). Our own calculations, utterly trivial by comparison, were done with Maple V. The paper Hodges [9] (cf. [10]) gave a compositional semantics for a language with some devices of imperfect information. The language was complicated, because it allowed imperfect information both at quantifiers and at conjunctions and disjunctions.