International Journal of Algebra and Computation

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron–Frobenius theory.

Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, b_D) be a Sylow B-subgroup of S(n). We consider the fusion system and determine a precise formula for its essential rank. In addition, the p-blocks B which admit a p-local subgroup of S(n) controlling B-fusion are characterized.

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut^⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by 2s for an orientable equation and by 12s⁴ for a non-orientable equation, where s is the sum of the lengths of the coefficients.

We describe a manageable set of relations that generates the finitary relational clone of an algebra with a parallelogram term. This result applies to any algebra with a Maltsev term and to any algebra with a near unanimity term. One consequence of the main result is that on any finite set and for any finite k there are only finitely many clones of algebras with a k-ary parallelogram term which generate residually small varieties.

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.