In this paper, the concepts of a weak Hopf algebra and a quasi-braided almost bialgebra are introduced. It is shown that the quantum quasi-doubles of some weak Hopf algebras are quasi-braided almost bialgebras. This fact implies that some new solutions of the quantum Yang-Baxter equation can be constructed from some weak Hopf algebras, in particular, when the weak Hopf algebra is a finite Clifford monoid algebra.

This work is a development of braids, tensor categories and Yang-Baxter operators. According to Li [Li, F. (1998). Weak Hopf algebras and some new solution of quantum Yang-Baxter equation. J. AIgebra 208:72-100; Li, F. (2000). Solutions of Yang-Baxter equation in endomorphism semigroup and quasi-(co)braided almost bialgebras. Comm. Algebra 28 (5): 2253-2270], it can be seen as a continuation of studying (not necessarily invertible) solutions of the (quantum) Yang-Baxter equation. We firstly introduce the right braid monoids and discuss their properties. Then, we define pre-tensor categories, pre-tensor functors and quasi-braided pre-tensor categories, and investiage their characterizations. Three examples are given from respectively a weak Hopf algebra, a crossed S-set of a Clifford monoid and the (strict) right braid category. Two universalities of the (strict) right braid category are gotten in order to characterize a category of general Yang-Baxter operators and a quasi-braided pre-tensor category. In a pretensor category we build a general centre of a pre-tensor category as a generalization of a centre and show that it is a quasi-braided pre-tensor category. At the end, a categorical interpretation of the quantum quasi-double of a weak Hapf algebra is obtained under a certain condition.

本文在余代数上定义了五类等价关系，它们是Green等价g，D，L，R，H。然后给出了这些等价关系一些基本性质和结构特点。在每个Green等价的等价类集上构造了一种偏序并给出了偏序上半格和结构的刻划。用g-，L-，R-类分别出了子余代数、左余理想、右余理想的结构刻划。进一步地，本文研究了张量积上的Green等价划及余代数同态的Green保持性和提升性。作为应用，本文得到了两个不可约余代数的张量积仍为不可约余代数的一人重要条件；证明了不可约余、代数在g-保持余代数同态下的同态像是不可约的。

In this paper, the author studies the regularity of Munn rings and completely 0-simple semigroup rings. When either (i) R is a ring with identity and I and Λ are infinite, or (ii) R is a strong IBN and fully Dedekind-finite ring with identity and either I or Λ is finite, in Section 2, the regularity of the Munn ring M (R; I, Λ; P) is characterized; in Section 3, for a completely 0-simple semigroup S=Mo (G; I, Λ; P), the regularity of RS is characterized. Meantime, the author shows that for a locally finite monoid S and a ring R with identity, RS is a strong IBN ring if and only if R is so; RS is fully Dedekind-finite if and only if R is so.

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.