Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^{?}$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.     

Morita equivalence of finite semigroups

Semigroup Forum - Two semigroups are called Morita equivalent if the categories of firm right acts over them are equivalent. We prove that every semigroup is Morita equivalent to its subsemigroup...     

Strong Morita equivalence for ordered semigroups with local units

We prove that partially ordered semigroups with local units are strongly Morita equivalent if and only if they have a joint enlargement, which in turn happens if and only if the Cauchy completions of the semigroups are equivalent.     

Elementary equivalence of incidence rings over semi-perfect rings

In this paper, we prove that if two incidence rings constructed by the same semiperfect ring and some two quasi-ordered sets are elementarily equivalent, then the given sets are elementarily equivalent.     

On deciding switching equivalence of graphs

We prove that the problem of deciding whether two graphs are switching equivalent is polynomial time equivalent to deciding isomorphism of graphs. The implications of this to the analysis of two-graphs are discussed.     

Measure equivalence rigidity and bi-exactness of groups

We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A?G, B?Γ of non-amenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of non-amenable exact direct product groups.     

On the equivalence of financial structures with long-term assets

In a stochastic financial exchange economy, two financial structures are equivalent if, for each given state price, the marketable payoffs are identical for the associated asset prices. The key property of two equivalent financial structures is that, when associated with any standard exchange economy, they lead to the same financial equilibrium. We exhibit a sufficient condition for the equivalence of two financial structures without re-trading with possibly long-term assets. We then apply this result to financial structures built upon primitive assets and their re-trading. We also borrow an assumption from Bonnisseau and Chéry (Ann Financ 10:523–552, 2014) to prove the equivalence between a financial structure and its reduced forms.     

Elementary equivalence of semigroups of invertible matrices with nonnegative elements over commutative partially ordered rings

In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements are elementarily equivalent.     

Typical equivalence of linear groups and other algebraic systems

The paper is devoted to the notion of typical equivalence introduced by B. I. Plotkin. We give some examples of elementarily equivalent objects that are not typically equivalent and show two ways to construct nonisomorphic typically equivalent algebras. We also prove A. I. Maltsev??s theorem on elementary equivalence of linear groups over fields for the case of typical equivalence.     

Composition of matrix products and categorical equivalence

First, we prove two finite algebras are categorically equivalent if and only if the matrix products of their irredundant non-refinable covers are isomorphic. Second, we characterize families of irreducible algebras such that there exists an algebra whose neighbourhoods in an irredundant non-refinable cover are isomorphic to the respective irreducible algebra in the given family. Finally, we exhibit two facts by constructing examples. The first one is that there is a family of irreducible algebras such that there are many algebraic structures whose neighbourhoods in an irredundant non-refinable cover are isomorphic to the respective irreducible algebra in the given family. The second example is an algebra such that the matrix product of an irredundant non-refinable cover is bigger than the given algebra.     

Quasisymmetric equivalence of self-similar sets

We prove that the self-similar sets with the strong separation condition are all quasisymmetrically equivalent.     

Classification of cocycles of generic equivalence relations

We study cocycles of an ergodic generic countable equivalence relation ℜ modulo meager sets. Two cocycles of ℜ are called weakly equivalent if they are cohomologous up to an element of Aut ℜ. It is proved that two nontransient cocycles with values in an arbitrary countable group are weakly equivalent if and only if their generic Mackey actions are isomorphic.     

Morita equivalence and Pedersen ideals

We show that two -algebras are strongly Morita equivalent if and only if their Pedersen ideals are Morita equivalent as rings with involution.

     

On the classification of all self-dual additive codes over GF(4) of length up to 12

We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these codes can be represented as graphs, and that two codes are equivalent if and only if the corresponding graphs are equivalent with respect to local complementation and graph isomorphism. We use these facts to classify all codes of length up to 12, where previously only all codes of length up to 9 were known. We also classify all extremal Type II codes of length 14. Finally, we find that the smallest Type I and Type II codes with trivial automorphism group have length 9 and 12, respectively.     

Elementary equivalence of the semigroup of invertible matrices with nonnegative elements

In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly ordered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 39–53, 2006.     

Witt equivalence of semilocal Dedekind domains in global fields

We examine a condition for two semilocal Dedekind rings, the fields of fractions of which are global fields, to be Witt equivalent. To solve the problem we generalize the notion of a Hilbert-symbol equivalence introduced in [11] and prove that a Witt equivalence is equivalent to a Hilbert-symbol equivalence. As a result we describe a Witt equivalence in terms of field invariants.     

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.     

Context equivalence of semigroups

Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.