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In 1998, Laan introduced weakly pullback flat acts and showed that weak pullback flatness is equivalent to the conjunction of Conditions (P) and (E′). In 2013, Golchin introduced Condition (P′), which is a generalization of Condition (P). In this article, we first define Condition (PF″), which lies strictly between weak pullback flatness and Condition (P′), and prove that Condition (PF″) coincides with the conjunction of Conditions (P′) and (E′). Furthermore, we give a classfication of monoids by Condition (PF″) of (cyclic, Rees factor) acts, and find a necessary and sufficient condition under which Condition (PF″) coincides with Condition (P′) (resp., weak pullback flatness, strong flatness) for Rees factor acts. Finally, we investigate Conditions (P′), (E′) and (PF″) using some new types of epimorphisms. [ABSTRACT FROM PUBLISHER]

We study problems inspired by language theory in certain matrix monoids which are extensions of free monoids. In particular, we characterize free two-element sets, morphisms and morphic equality sets in these monoids. [ABSTRACT FROM AUTHOR]

We explore some properties of Schreier split epimorphisms between monoids, which correspond to monoid actions. In particular, we prove that the split short five lemma holds for monoids, when it is restricted to Schreier split epimorphisms, and that any Schreier reflexive relation is transitive, partially recovering in monoids a classical property of Mal'tsev varieties. [ABSTRACT FROM AUTHOR]

Abstract: We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg–Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. We explain at the end of the paper that a similar phenomenon occurs in many other situations. [Copyright &y& Elsevier]

Abstract: We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. The duoidal categories (introduced by Aguiar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not ‘classical.’ In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include the -category of 2-categories, 2-functors and pseudonatural transformations and Tamarkin’s homotopy 2-category of -categories, -functors and coherent -transformations. [Copyright &y& Elsevier]

Abstract: Length preserving morphisms and inverse of substitutions are two well-studied operations on regular languages. Their connection with varieties generated by power monoids was established independently by Reutenauer and Straubing in 1979. More recently, an ordered version of this theory was proposed by Polák and by the authors. In this paper, we present an improved version of these results and obtain the following consequences. Given a variety of finite ordered monoids , let be the variety of finite ordered monoids generated by the upper set monoids of members of . Then . This contrasts with the known results for the unordered case: the operator corresponding to power monoids satisfies , but the varieties , , and can be distinct. [Copyright &y& Elsevier]

The aim of this work is to study monoid morphisms between commutative monoids. Algorithms to check if a monoid morphism between two finitely generated monoids is injective and/or surjective are given. The structure of the set of monoid morphisms between a monoid and a cancellative monoid is also studied and an algorithm to obtain a system of generators of this set is provided. [ABSTRACT FROM AUTHOR]

Let $ M$-dimensional smooth manifold and $ X$ $ \mathbb{H}_*(LM):=H_{*+d}(LM)$, $ H_*(\Omega^2 X)$ be a topological monoid with a homotopy inverse. Suppose that $ G$. We define a structure of Batalin-Vilkovisky algebra on $ H_*(\Omega^2BG)\otimes\mathbb{H}_*(M)$ $ H_*(\Omega^2BG)$

Abstract: This paper presents non-commutative and structural notions of torsor. The two are related by the machinery of Tannaka–Krein duality. [Copyright &y& Elsevier]

Abstract: This paper studies the ambiguity of morphisms in free monoids. A morphism is said to be ambiguous with respect to a string if there exists a morphism which differs from for a symbol occurring in , but nevertheless satisfies ; if there is no such then is called unambiguous. Motivated by the recent initial paper on the ambiguity of morphisms, we introduce the definition of a so-called segmented morphism , which, for any , maps every symbol in an infinite alphabet onto a word that consists of distinct factors in , where and are different letters. For every , we consider the set of those finite strings over an infinite alphabet with respect to which is unambiguous, and we comprehensively describe its relation to any , . Thus, our work features the first approach to a characterisation of sets of strings with respect to which certain fixed morphisms are unambiguous, and it leads to fairly counter-intuitive insights into the relations between such sets. Furthermore, it shows that, among the widely used homogeneous morphisms, most segmented morphisms are optimal in terms of being unambiguous for a preferably large set of strings. Finally, our paper yields several major improvements of crucial techniques previously used for research on the ambiguity of morphisms. [Copyright &y& Elsevier]