Narrow Results

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids.

For a monoid $M$ and a class of partial actions of $M$, determined by a set, $R$, of identities, we define ${{ℱ}{ℛ}}_R(M)$ to be the universal $M$-generated two-sided restriction monoid with respect to partial actions of $M$ determined by $R$. This is an $F$-restriction monoid which (for a certain $R$) generalizes the Birget–Rhodes prefix expansion ${\tilde{G}}^{{ℛ}}$ of a group $G$. Our main result provides a coordinatization of ${{ℱ}{ℛ}}_R(M)$ via a partial action product of the idempotent semilattice $E({{ℱ}{ℐ}}_R(M))$ of a similarly defined inverse monoid, partially acted upon by $M$. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result.

We show that some properties of ${{ℱ}{ℛ}}_R(M)$ agree well with suitable properties of $M$, such as being cancellative or embeddable into a group. We observe that if $M$ is an inverse monoid, then ${{ℱ}{ℐ}}_{R_s}(M)$, the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion $M^{\mathrm{\text{pr}}}$. This gives a presentation of $M^{\mathrm{\text{pr}}}$ and leads to a model for ${{ℱ}{ℛ}}_{R_s}(M)$ in terms of the known model for $M^{\mathrm{\text{pr}}}$.

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra $A$ on an algebra $R$, it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math.537 (2011) 13–30], have shown that this is always possible if $R$ is unital. We are interested in investigating the situation, where both algebras $A$ and $R$ are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc.342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc.135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

In this paper, we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.

In this paper, we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring. We present sufficient conditions under which the partial skew groupoid ring is either associative or unital. Also, we show that there is a one-to-one correspondence between partial actions of an ordered groupoid G on a ring R, in which the domain of each partial bijection is an ideal, and meet-preserving global actions of the Birget–Rhodes expansion G^BR of G on R. Using this correspondence, we prove that the partial skew groupoid ring is a homomorphic image of the skew groupoid ring constructed through the Birget–Rhodes expansion.

In this paper we study partial actions of abelian groups on a ring R having an enveloping action. Among other results we prove that if α is such a partial action and (T, β) its globalization, then there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R ⋆_α G and closed (respectively, T-disjoint prime) ideals of T ⋆_β G. We also prove that there exists a one-to-one correspondence between closed (respectively, R-disjoint prime) ideals of R ⋆_α G and closed (respectively, -disjoint prime) ideals of , where is the left α-quotient ring of R. Finally, we use these results to study strongly prime ideals and nonsingular prime ideals of R ⋆_α G.

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra $A$, and show that this coring is Galois if and only if $A$ is an $\alpha$-partial Galois extension of its coinvariants.

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#Â$. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.

In this paper, we introduce the concept of a $\lambda$-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a $\lambda$-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.

In the paper, we discuss the partial actions of Sweedler Hopf algebra on the generalized quaternion algebra, and give the sufficient and necessary conditions to make the actions be the partial actions. As an application, we give a complete description of all partial actions of Sweedler Hopf algebra on the generalized quaternion algebra.