Chain varieties of monoids

A variety of universal algebras is called a chain variety if its subvariety lattice is a chain. Non-group chain varieties of semigroups were completely classified by Sukhanov in 1982. Here we completely determine non-group chain varieties of monoids as algebras of tyoe (2,0).Comment: 76 pages, 3 figures, 3 tables. In comparison with the previous version, we made a number of linguistic corrections onl

Chain Equivalences for Symplectic Bases, Quadratic Forms and Tensor Products of Quaternion Algebras

We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in characteristic 2 is an immediate result. Based on the chain equivalences for quadratic forms, both in characteristic 2 and not 2, we provide chain equivalences for tensor products of quaternion algebras over fields with no nontrivial 3-fold Pfister forms. The chain equivalence for biquaternion algebras in characteristic 2 is also obtained in this process, without any assumption on the base-field.Comment: 10 page

Homotopy Diagrams of Algebras

In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant operads and on the principle that algebras over cofibrant operads are homotopy invariant. In our approach, algebraic models for colored operads describing diagrams of homomorphisms played an important role. The aim of this paper is to give an explicit description of these models. A possible application is an appropriate formulation of the `ideal' homological perturbation lemma for chain complexes with algebraic structures. Our results also provide a conceptual approach to `homotopies through homomorphism' for strongly homotopy algebras. We also argue that strongly homotopy algebras form a honest (not only weak Kan) category. The paper is a continuation of our program to translate the famous book "M. Boardman, R. Vogt: Homotopy Invariant Algebraic Structures on Topological Spaces" to algebra.Comment: 24 pages, LaTeX 2.0