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The construction of Sweedler duality is an important tool in the theory of Hopf algebras over a field, which is a right adjoint to the dual algebra functor. In this paper, we study the Sweedler duality of Hom-algebras and their Hom-modules. We delve into the structure of Hom-coalgebras and derive the linear morphisms associated with them. Additionally, as an application, we present the (right) Hom-(co)module morphisms under the Sweedler duality., Comment: 11 pages

Leibniz algebras can be seen as a ``non-commutative" analogue of Lie algebras. Nijenhuis operators on Leibniz algebras introduced by Cari\~{n}ena, Grabowski, and Marmo in [J. Phys. A: Math. Gen. 37(2004)] are (1, 1)-tensors with vanishing Nijenhuis torsion. Recently triangular Leibniz bialgebras were introduced by Tang and Sheng in [J. Noncommut. Geom. 16(2022)] via the twisting theory of twilled Leibniz algebras. In this paper we find that Leibniz algebras are very closely related to Nijenhuis operators, and prove that a triangular symplectic Leibniz bialgebra together with a dual triangular structure must possess Nijenhuis operators, which makes it possible to study the applications of Nijehhuis operators from the perspective of Leibniz algebras. At the same time, we regain the classical Leibniz Yang-Baxter equation by using the tensor form of classical $r$-matrics. At last we give the classification of triangular Leibniz bialgebras of low dimensions.

The Drinfeld double associated to the weak multiplier Hopf ($*$-) algebra pairing $\left\langle A, B\right\rangle$ is constructed. We show that the Drinfeld double is again a weak multiplier Hopf ($*$-) algebra. If $A$ and $B$ are algebraic quantum groupoids, then so does the double. We also prove the correspondence between modules over the Drinfeld double and Yetter-Drinfeld modules. Finally, we prove that the double is a quasitriangular weak multiplier Hopf algebra.

Let $H$ be a bialgebra. Let $\sigma: H\otimes H\to A$ be a linear map, where $A$ is a left $H$-comodule coalgebra, and an algebra with a left $H$-weak action $\triangleright$. Let $\tau: H\otimes H\to B$ be a linear map, where $B$ is a right $H$-comodule coalgebra, and an algebra with a right $H$-weak action $\triangleleft$. In this paper, we improve the necessary conditions for the two-sided crossed product algebra $A\#^{\sigma} H~{^{\tau}\#} B$ and the two-sided smash coproduct coalgebra $A\times H\times B$ to form a bialgebra (called double crossed biproduct) such that the condition $b_{[1]}\triangleright a_0\otimes b_{[0]}\triangleleft a_{-1}=a\otimes b$ in Majid's double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzez\'nski's crossed product and give some applications., Comment: Communications in Algebra,2022

We may give rise to some questions related to the mathematical structures of $P$-class and $NP$-class. We have seen that one is a proper subclass of the other. Here we disclose more that $P$- class turns out to be the proper distributive sublattice of the $NP$- class., Comment: arXiv admin note: substantial text overlap with arXiv:2003.05321, arXiv:1912.10849

We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 and 4 are given. At last we prove that Rota-Baxter Poisson T-algebras can produce pre-Poisson T-algebras and Poisson T-algebras can be obtained from pre-Poisson T-algebras.

We introduce the notion of a four-angle $H$-Hopf module for a Hom-Hopf algebra $(H,\beta)$ and show that the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ of four-angle $H$-Hopf modules is a monoidal category with either a Hom-tensor product $\otimes_{H}$ or a Hom-cotensor product $\Box_{H}$ as a monoidal product. We study the category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules with bijective structure map can be organized as a braided monoidal category, in which we use a new monoidal structure and prove that if the canonical braiding of the category $\mathcal{YD}^{H}_{H}$ is symmetry then $(H,\beta)$ is trivial. We then prove an equivalence between the monoidal category $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ or $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$ of four-angle $H$-Hopf modules, and the monoidal category $\mathcal{YD}^{H}_{H}$ of Yetter-Drinfel'd modules, and furthermore, we give a braiding structure of the monoidal categorys $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\otimes_{H})$ (and $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$). Finally, we prove that when $(H,\beta)$ is finite dimensional Hom-Hopf algebra, the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ is isomorphic to the representation category of Heisenberg double $H^{*op}\otimes H^{*}\#H\otimes H^{op}$.

Let $(H, \beta)$ be a monoidal Hom-Hopf algebra with the bijective antipode $S$, In this paper, we mainly construct the Drinfel'd codouble $T(H)=(H^{op}\otimes H^{*}, \beta\otimes \beta^{*-1})$ and $\widehat{T(H)}=( H^{*}\otimes H^{op}, \beta^{*-1}\otimes \beta)$ in the setting of monoidal Hom-Hopf algebras. Then we prove both $T(H)$ and $\widehat{T(H)}$ are coquasitriangular. Finally, we discuss the relation between Drinfel'd codouble and Heisenberg double in the setting of monoidal Hom-Hopf algebras, which is a generalization of the part results in \cite{L94}.

Let $H$ be a Hopf algebra and $\mathcal{LR}(H)$ the category of Yetter-Drinfel'd-Long bimodules over $H$. We first give sufficient and necessary conditions for $\mathcal{LR}(H)$ to be symmetry and pseudosymmetry, respectively. We then introduce the definition of $u$-condition in $\mathcal{LR}(H)$ and discuss the relation between the $u$-condition and the symmetry of $\mathcal{LR}(H)$. Finally, we show that $\mathcal{LR}(H)$ over a triangular (cotriangular, resp.) Hopf algebra contains a rich symmetric subcategory.

In this paper, we first discuss some properties of the Galois linear maps. We provide some equivalent conditions for Hopf algebras and Hopf (co)quasigroups as its applications. Then let $H$ be a Hopf quasigroup with bijective antipode and $G$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a new category $\mathscr{C}_{H}(\alpha,\beta)$ with $\alpha,\beta\in G$ over $H$ and construct a new braided $\pi$-category $\mathscr{C}(H)$ with all the categories $\mathscr{C}_{H}(\alpha,\beta)$ as components., Comment: 16 pages

Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $\pi-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the construction method and fundamental theorem of them. Thirdly, we research the smash products of $Q$-graded Hopf quasigroups., Comment: $Q$-graded Hopf quasigroups; $Q$-graded Hopf quasimodules; Smash products

Let $H$ be a Hopf quasigroup with bijective antipode and let $Aut_{HQG}(H)$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a category ${_{H}\mathcal{YDQ}^{H}}(\alpha,\beta)$ with $\alpha,\beta\in Aut_{HQG}(H)$ and construct a braided $T$-category $\mathcal{YDQ}(H)$ having all the categories ${_{H}\mathcal{YDQ}^{H}}(\alpha,\beta)$ as components., Comment: arXiv admin note: text overlap with arXiv:1502.07377

In this paper we study twisted algebras of multiplier Hopf ($^*$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O]., Comment: 32. arXiv admin note: text overlap with arXiv:math/0504386 by other authors

Let $(A,\Delta)$ be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. The main difference with multiplier Hopf algebras is that now, the canonical maps $T_1$ and $T_2$ on $A\otimes A$, defined by $$T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\quad\text{and}\qquad\quad T_2(c\otimes a)=(c\otimes 1)\Delta(a),$$ are no longer assumed to be bijective. Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from $A$ to itself. In this paper, we introduce and study the notion of integrals on such regular weak multiplier Hopf algebras. A left integral is a non-zero linear functional on $A$ that is left invariant (in an appropriate sense). Similarly for a right integral. For a regular weak multiplier Hopf algebra $(A,\Delta)$ with (sufficiently many) integrals, we construct the dual $(\widehat A,\widehat\Delta)$. It is again a regular weak multiplier Hopf algebra with (sufficiently many) integrals. This duality extends the known duality of finite-dimensional weak Hopf algebras to this more general case. It also extends the duality of multiplier Hopf algebras with integrals, the so-called algebraic quantum groups. For this reason, we will sometimes call a regular weak multiplier Hopf algebra with enough integrals an algebraic quantum groupoid. We discuss the relation of our work with the work on duality for algebraic quantum groupoids by Timmermann. We also illustrate this duality with a particular example in a separate paper. In this paper, we only mention the main definitions and results for this example. However, we do consider the two natural weak multiplier Hopf algebras associated with a groupoid in detail and show that they are dual to each other in the sense of the above duality.

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by A. Van Daele and S. H. Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids., Comment: arXiv admin note: text overlap with arXiv:1306.1466 by other authors

Let $H$ be a finite dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra $H\o H^*$ as monoidal category. Moreover if $H$ is a Hopf algebra with bijective antipode, the isomorphism is braided., Comment: to appear in Czechoslovak Mathematical Journal

Let $\pi$ be a group. The aim of this paper is to construct the category of Yetter-Drinfeld modules over the quasi-Turaev group coalgebra $H=(\{H_\a\}_{\a\in\pi},\Delta,\varepsilon,S,\Phi)$, and prove that this category is isomorphic to the center of the representation category of $H$. Therefore a new Turaev braided group category is constructed.

In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. If $H$ is a weak monoidal Hom-Hopf algebra with bijective antipode and let $Aut_{wmHH}(H)$ be the set of all automorphisms of $H$. Then we introduce a category ${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ with $\alpha,\beta\in Aut_{wmHH}(H)$ and construct a braided $T$-category $\mathcal{WMHYD}(H)$ that having all the categories ${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ as components., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:1405.6767

Let $ Aut_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{MHD}(H)$ and prove that the $T$-category $Rep(\mathcal{MHD}(H))$ of representation of $\mathcal{MHD}(H)$ is isomorphic to $\mathcal {MHYD}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $\mathcal{ZMHYD}(H)$ over $\mathbb{Z},$ generalizing the main construction by Staic \cite{S2007}., Comment: arXiv admin note: substantial text overlap with arXiv:1405.6767

Let $(A,\alpha)$ be a finite-dimensional Hom-Hopf algebra. In this paper we mainly construct the Drinfel'd double $D(A)=(A^{op}\bowtie A^{\ast},\alpha\otimes(\alpha^{-1})^{\ast})$ in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see \cite{M2}) and another one is to introduce the notion of dual pairs of of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double $D(A)$ and Heisenberg double $H(A)=A\# A^{*}$, generalizing the main result in \cite{Lu}. Especially, the examples given in the paper are not obtained from the usual Hopf algebras.

Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal MHYD}(H)=\{{}_{H}\mathcal {MHYD}^{H}(A, B)\}_{(A, B)\in G}$ forms a braided $T$-category, generalizing the main constructions construction by Panaite and Staic (Isr J Math 158:349-365, 2007).

In this paper, we continue the study of weak multiplier Hopf algebras. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\otimes C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a separability idempotent. We also consider special cases and examples in this paper. In particular, we see how for any weak multiplier Hopf algebra, it is possible to make $C\otimes B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\otimes C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

A weak multiplier Hopf algebra is a pair (A,\Delta) of a non-degenerate idempotent algebra A and a coproduct $\Delta$ on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra M(A\otimes A) with some natural extra properties (like the existence of a counit). Further we impose extra but natural conditions on the ranges and the kernels of the canonical maps T_1 and T_2 defined from A\otimes A to M(A\otimes A) by T_1(a\otimes b)=\Delta(a)(1\otimes b) and T_2(a\ot b)=(a\otimes 1)\Delta(b). The first condition is about the ranges of these maps. It is assumed that there exists an idempotent element E\in M(A\otimes A) such that \Delta(A)(1\ot A)=E(A\ot A) and (A\otimes 1)\Delta(A)=(A\otimes A)E. The second condition determines the behavior of the coproduct on the legs of E. We require (\Delta\otimes \iota)(E)=(\iota\otimes\Delta)(E)=(1\otimes E)(E\ot 1)=(E\otimes 1)(1\otimes E) where $\iota$ is the identity map and where $\Delta\otimes \iota$ and $\iota\otimes\Delta$ are extensions to the multipier algebra M(A\otimes A). Finally, the last condition determines the kernels of the canonical maps T_1 and T_2 in terms of this idempotent E by a very specific relation. From these conditions we develop the theory. In particular, we construct a unique antipode satisfying the expected properties and various other data. Special attention is given to the regular case (that is when the antipode is bijective) and the case of a *-algebra (where regularity is automatic). Weak Hopf algebras are special cases of such weak multiplier Hopf algebras. Conversely, if the underlying algebra of a (regular) weak multiplier Hopf algebra has an identity, it is a weak Hopf algebra. Also any groupoid, finite or not, yields two weak multiplier Hopf algebras in duality.

Let $G$ be a {\it finite group}. Consider the algebra $A$ of all complex functions on G (with pointwise product). Define a coproduct $\Delta$ on A by $\Delta(f)(p,q)=f(pq)$ where $f\in A$ and $p,q\in G$. Then $(A,\Delta)$ is a Hopf algebra. If $G$ is only a {\it groupoid}, so that the product of two elements is not always defined, one still can consider $A$ and define $\Delta(f)(p,q)$ as above when $pq$ is defined. If we let $\Delta(f)(p,q)=0$ otherwise, we still get a coproduct on $A$, but $\Delta(1)$ will no longer be the identity in $A\ot A$. The pair $(A,\Delta)$ is not a Hopf algebra but a weak Hopf algebra. If $G$ is a {\it group}, but {\it no longer finite}, one takes for $A$ the algebra of functions with finite support. Then $A$ has no identity and $(A,\Delta)$ is not a Hopf algebra but a multiplier Hopf algebra. Finally, if $G$ is a {\it groupoid}, but {\it not necessarily finite}, the standard construction above, will give, what we call in this paper, a weak multiplier Hopf algebra. Indeed, this paper is devoted to the development of this 'missing link': {\it weak multiplier Hopf algebras}. We spend a great part of this paper to the motivation of our notion and to explain where the various assumptions come from. The goal is to obtain a good definition of a weak multiplier Hopf algebra. Throughout the paper, we consider the basic examples and use them, as far as this is possible, to illustrate what we do. In particular, we think of the finite-dimensional weak Hopf algebras. On the other hand however, we are also inspired by the far more complicated existing analytical theory. In forthcoming papers on the subject, we develop the theory further.

Let $A$ and $B$ be two algebraic quantum groups (i.e. multiplier Hopf algebras with integrals). Assume that $B$ is a right $A$-module algebra and that $A$ is a left $B$-comodule coalgebra. If the action and coaction are matched, it is possible to define a coproduct $\Delta_#$ on the smash product $A # B$ making the pair $(A # B,\Delta_#)$ into an algebraic quantum group. In this paper, we continue the study of these objects. First, we study the various data of the bicrossproduct $A # B$, such as the modular automorphisms, the modular elements, ... and obtain formulas in terms of the data of the components $A$ and $B$. Secondly, we look at the dual of $A # B$ (in the sense of algebraic quantum groups) and we show it is itself a bicrossproduct (of the second type) of the duals $\hatA$ and $\hatB$. The result is immediate for finite-dimensional Hopf algebras and therefore it is expected also for algebraic quantum groups. However, it turns out that some aspects involve a careful argument, mainly due to the fact that coproducts and coactions have ranges in the multiplier algebras of the tensor products and not in the tensor product itself. Finally, we also treat some examples in this paper. We have included some of the examples that are known for finite-dimensional Hopf algebras and show how they can also be obtained for more general algebraic quantum groups. We also give some examples that are more typical for algebraic quantum groups. In particular, we focus on the extra structure, provided by the integrals and associated objects. It should be mentioned that with examples of bicrossproducts of algebraic quantum groups, we do get examples that are essentially different from those commonly known in Hopf algebra theory.

Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ is always semisolvable. That is, such Hopf algebras can be obtained by (a number of) extensions from group algebras or duals of group algebras., Comment: The title has been changed, and some minor typos have been corrected. arXiv admin note: substantial text overlap with arXiv:1105.4398; and text overlap with arXiv:math/0305038 by other authors

We propose a definition of multiplier infinitesimal bialgebra and a definition of derivator Lie bialgebra. We give some examples of these structures and prove that every bibalanced multiplier infinitesimal bialgebra gives rise to a multiplier Lie bialgebra., Comment: 11 pages

In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A # B. The left coaction of B on A gives a possible coproduct on A # B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for this to be a proper coproduct on A # B. The result is again a regular multiplier Hopf algebra. Majid's construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair (C,D) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A # B and the smash product C # D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The *-algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G=KH and the intersection of H and K is trivial will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention.

In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formulas in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of Hopf algebras and work with multiplier Hopf algebras. Moreover, whereas in the finite-dimensional case, there is a complete symmetry between the bialgebra and its dual, this is no longer the case in infinite dimensions. Therefore we consider a direct version (with integrals) and a dual version (with cointegrals) of the Larson-Sweedler theorem. We also add some results about the antipode. Furthermore, in the process of this paper, we obtain a new approach to multiplier Hopf algebras with integrals.

We put the known results on the antipode of a usual quasitriangular Hopf algebra into the framework of multiplier Hopf algebras. We illustrate with examples which can not be obtained using classical Hopf Algebras. The focus of the present paper lies on the class of the so-called group cograded multiplier Hopf algebras. By doing so, we bring the results of quasitriangular Hopf group-coalgebras (as introduced by Turaev) to the more general framework of multiplier Hopf algebras.

Using the theory of corings, we generalize and unify Morita contexts introduced by Chase and Sweedler, Doi, and Cohen, Fischman and Montgomery. We discuss when the contexts are strict. We apply our theory corings arising from entwining structures, and this leads us to the notion of cleft entwining structure., Comment: 27 pages