This paper lays the foundations of a combinatorial homotopy theory, called A-theory, for simplicial complexes, which reflects their connectivity properties. A collection of bigraded groups is constructed, and methods for computation are given. A Seifert-Van Kampen type theorem and a long exact sequence of relative A-groups are derived. A related theory for graphs is constructed as well. This theory provides a general framework encompassing homotopy methods used to prove connectivity results about buildings, graphs, and matroids.

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In this paper we consider a free associative algebra on three generators over an arbitrary field K. Given a term ordering on the commutative polynomial ring on three variables over K, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such more general monomial orderings in non-commutative Gröbner basis theory.

Abstract. We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations-namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2-and on the Ufnarovskii graph Γ(A). The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph Γ(A) to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.