In this paper we shall present a new constructron of the free inverse monoid on a set A’. Contrary to the prevrous constructions of 19, 111, our construction is symmetric and originates from classrcal ideas of language theory. The mgredlents of this construction are the free group on X and the relatron that associates to a word II’ of the free monoid on X, the set of all pairs (u, v) such that ...

In this article we study percolation on the Cayley graph of a free product of groups. Such a graph has a tree-like structure which allows us to evaluate the critical values of the phase transition, mean cluster size and the critical exponent in bond percolation.

Associated to a finite graph X is its quantum automorphism group G(X). We prove a formula of type G(X ∗ Y ) = G(X) ∗w G(Y ), where ∗w is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula μ(G ∗w H) = μ(G) ⊠ μ(H), where μ is the associated spectral measure. This is verified in two situations: one using free probability techniques, t...

Let G = (V,E) be a graph. For a non-empty subset of vertices S ⊆ V , and vertex v ∈ V , let δS(v) = |{u ∈ S : uv ∈ E}| denote the cardinality of the set of neighbors of v in S, and let S = V − S. Consider the following condition: δS(v) ≥ δS(v) + k, (1) which states that a vertex v has at least k more neighbors in S than it has in S. A set S ⊆ V that satisfies Condition (1) for every vertex v ∈ ...

In this paper we prove that the derivability problems for productfree Lambek calculus and product-free Lambek calculus allowing empty premises are NP-complete. Also we introduce a new derivability characterization for these calculi.