Narrow Results

We study operad with a prescribed element (of its underlying degree one component), generalize some of the classical results of Gerstenhaber and Voronov [M. Gerstenhaber and A. A. Voronov, Homotopy G-algebras and moduli space operad, Int. Math. Res. Not.3 (1995) 141–153]. In particular, we introduce and show that Hom–Loday algebra cohomologies carry such structures when the Hom-structure twist is an idempotent.

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

We present a deformation theory associated to the higher Hochschild cohomology $H_{S^2}^{\ast }(A,A)$. We also study a $G$-algebra structure associated to this deformation theory.

Preclones are described as the closed classes of the Galois connection induced by a preservation relation between operations and matrix collections. The Galois closed classes of matrix collections are also described by explicit closure conditions.

In this paper, we study one class of commutative operads, namely, the operads of multidimensional (hollow) cubes in Euclidean spaces and their generalization. We describe the varieties of universal algebras rational equivalent to the varieties of algebras over commutative operads.

This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.

In this survey article, we report some new results of Gröbner-Shirshov bases, including new Composition-Diamond lemmas and some applications of some known Composition-Diamond lemmas.

We briefly summarize recent constructions of the disuccessor and duplicator of a quadratic operad and relate them to the Manin products, and Rota-Baxter and average operators.