scholarly journals Tensor products of finitely presented functors

2021 ◽  
pp. 2250186
Author(s):  
Martin Bies ◽  
Sebastian Posur
Keyword(s):  
Tensor Products ◽  
Main Tool ◽  
Universal Property ◽  
Unified Approach ◽  
Functor Categories ◽  
Explicit Constructions ◽  
Monoidal Structure ◽  
Proof Of Correctness ◽  
Finitely Presented

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.

scholarly journals TENSOR PRODUCTS OF STEINBERG ALGEBRAS

2019 ◽  
pp. 1-16 ◽  
Author(s):  
SIMON W. RIGBY
Keyword(s):  
Tensor Products ◽  
Isomorphism Problem ◽  
Universal Property ◽  
Easy Proof

We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras. We then consider the isomorphism problem for tensor products of Leavitt algebras, and show that no diagonal-preserving isomorphism exists between $L_{2,R}\otimes L_{3,R}$ and $L_{2,R}\otimes L_{2,R}$ . In fact, there are no unexpected diagonal-preserving isomorphisms between tensor products of finitely many Leavitt algebras. We give an easy proof that every $\ast$ -isomorphism of Steinberg algebras over the integers preserves the diagonal, and it follows that $L_{2,\mathbb{Z}}\otimes L_{3,\mathbb{Z}}\not \cong L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ (as $\ast$ -rings).

Compound Invariants and Mixed F-, DF-Power Spaces

1998 ◽  
Vol 50 (6) ◽  
pp. 1138-1162 ◽  
Author(s):  
P. A. Chalov ◽  
T. Terzioğlu ◽  
V. P. Zahariuta
Keyword(s):  
Power Series ◽  
Tensor Products ◽  
Main Tool ◽  
Proper Definition ◽  
Definition Of ◽  
Image Position ◽  
Isomorphic Classification

AbstractThe problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed F-, DF-power series spaces, i.e. the spaces of the following kind where ai (p, q) = exp((p - λiq)ai), p,q ∈ ℕ, and λ = (λi)i∈ℕ, a = (ai)i∈ℕ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of F- and DF-types, respectively. The mrectangle characteristic of the space G(λ a) is defined as the number of members of the sequence (ïiÒ ai)i2N which are contained in the union of m rectangles Pk = (δk, εk] ✗ (τk, tk], k = 1, 2 , . . . , m. It is shown that each m-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).

ON THE WORD PROBLEM FOR TENSOR PRODUCTS AND AMALGAMS OF MONOIDS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 271-294 ◽  
Author(s):  
JEAN-CAMILLE BIRGET ◽  
STUART W. MARGOLIS ◽  
JOHN MEAKIN
Keyword(s):  
Word Problem ◽  
Linear Time ◽  
Tensor Products ◽  
Commutative Rings ◽  
Semigroup Rings ◽  
Similar Idea ◽  
Free Product With Amalgamation ◽  
Equality Problem ◽  
Pushdown Automata ◽  
Finitely Presented

We prove that the word problem for the free product with amalgamation S *UT of monoids can be undecidable, even when S and T are finitely presented monoids with word problems that are decidable in linear time, the factorization problems for U in each of S and T, as well as other problems, are decidable in polynomial time, and U is a free finitely generated unitary submonoid of both S and T. This is proved by showing that the equality problem for the tensor product S ⊗UT is undecidable and using known connections between tensor products and amalgams. We obtain similar results for semigroups, and by passing to semigroup rings, we obtain similar results for rings as well. The proof shows how to simulate an arbitrary Turing machine as a communicating pair of two deterministic pushdown automata and is of independent interest. A similar idea is used in a paper by E. Bach to show undecidability of the tensor equality problem for modules over commutative rings.

A Unified Approach to Study the Thermal Dynamics in Multilongitudinal Mode Semiconductor Lasers

2001 ◽  
Vol 20 (2) ◽  
pp. 159-169 ◽  
Author(s):  
M. Ganesh Madhan ◽  
P. R. Vaya ◽  
N. Gunasekaran
Keyword(s):  
Semiconductor Lasers ◽  
Unified Approach ◽  
Thermal Dynamics
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