We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with recently introduced \cite{AGV1} bi/Hopf-algebras that are among all support equivalent algebras. Our approach uses vector spaces endowed family of linear maps between tensor powers $A$, called $\Omega$-algebras. This allows us to treat algebras, coalgebras, braided many oth...

We investigate the representation and complete classes for algebras of partial functions with signature relative complement domain restriction. provide prove correctness a finite equational axiomatisation class representable by functions. As corollary, same equations axiomatise injective For representations, we show that is meet if only it join complete. Then completely precisely atomic algebra...

According to a result by K. B. Lee, the lattice of varieties of pseudocomplemented distributive lattices is the ui + 1 chain B_i C Bo C Bi C • ■ ■ C Bn C •■ • C Bw in which the first three varieties are formed by trivial, Boolean, and Stone algebras respectively. In the present paper it is shown that any Stone algebra is determined within Bi by its endomorphism monoid, and that there are at mos...

The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal enveloping algebra of a finite dimensional Lie algebra g with a finite subgroup G of automorphisms of g is Calabi-Yau if and only if the universal en...

Recently, a number of new classes of infinite-dimensional simple Lie algebras over a Field of characteristic 0 were discovered by several authors (see the references at the end of this paper). Among those algebras, are the generalized Witt algebras. The higher rank Virasoro algebras was introduced by Patera and Zassenhaus [PZ], which are 1-dimensional universal central extensions of some genera...

Two different notions of universal, one due to Hedrlín and Pultr in the 1960s and the other due to Sapir in the 1980s, are discussed, as well as the relationship between them. Some of the historical perspective and mathematical motivation lying behind them is also included, together with a brief overview of a variety meriting further investigation in this context.