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Register a new userTrace- and pseudo-products: restriction-like semigroups with a band of projections
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We ascertain conditions and structures on categories and semigroups which admit the construction of pseudo-products and trace products respectively, making their connection as precise as possible. This topic is modelled on the ESN Theorem and its generalization to ample semigroups. Unlike some other variants of ESN, it is self-dual (two-sided), and the condition of commuting projections is relaxed. The condition that projections form a band (are closed under multiplication) is shown to be a very natural one. One-sided reducts are considered, and compared to (generalized) D-semigroups. Finally the special case when the category is a groupoid is examined.
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When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ${}^{*}$-semigroups and commutative involution semigroups.
The concept of a k-translatable groupoid is explored in depth. Some properties of idempotent k-translatable groupoids, left cancellative k-translatable groupoids and left unitary k-translatable groupoids are proved. Necessary and sufficient conditions are found for a left cancellative k-translatable groupoid to be a semigroup. Any such semigroup is proved to be left unitary and a union of disjoint copies of cyclic groups of the same order. Methods of constructing k-translatable semigroups that are not left cancellative are given.
Pairwise non-isomorphic semigroups obtained from the finite inverse symmetric semigroup $mathcal{IS}_n ,$ finite symmetric semigroup $mathcal{T}_n$ and bicyclic semigroup by the deformed multiplication proposed by Ljapin are classified.
Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of A-infinity algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.
For each subchain $X$ of a chain $X$, let $T_{RE}(X, X)$ denote the semigroup under composition of all full regressive transformations, $alpha:Xrightarrow X$ satisfying $xalphaleq x$ for all $xin X$. Necessary and sufficient conditions for $T_{RE}(X,X)$ and $T_{RE}(Y,Y)$ to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations $T_{RE}(X,X)$ where $X$ are familiar subchains of $R$, the chain of real numbers.
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