‎‎The equivalence relation isoclinism partitions the class of all pairs of groups‎ ‎into families‎. ‎In this paper‎, ‎a complete classification of the set of all pairs $(G,G')$ is established‏‎,‎ whenever‎ $G$ is a $p$-group of order at most $p^5‎‏$‎ and ‎$p$ is a prime number greater than 3. Moreover‎, ‎the classification of pairs $(H,H')$ for extra special $p$-groups $H$ is also given.

In this note we introduce a new equivalence relation θ∗ on a (semi)hyperring R and we show that it is strongly regular. Also we prove that, R/θ∗, the equivalence class of this equivalence relation under usual operations consists a commutative (semi-)ring. Finally we introduce the notion of θ-parts of hyperrings and investigate the important properties of them. Mathematics Subject Classification...

In this work we consider modeling of workflow systems with Petri nets. The problem of replacing some resources by others retaining the correct functioning of the system is studied. The relation of resource s-equivalence is defined. Some properties of this relation are studied. It is proved that s-equivalence is decidable provided the set of all minimal sound resources for a given control state ...

We consider the homeomorphism problem for countable topological spaces and investigate its descriptive complexity as an equivalence relation. It is shown that even for countable metric spaces the homeomorphism problem is strictly more complicated than the isomorphism problem for countable graphs and indeed it is not Borel reducible to any orbit equivalence relation induced by a Borel action of ...

In this paper we define an equivalence relation on the set of positive irrational numbers less than 1. The relation is defined by means of continued fractions. Equivalence classes under this relation are determined by the places of some elements equal to 1 (called essential 1’s) in the continued fraction expansion of numbers. Analysis of suprema of all equivalence classes leads to a solution wh...

ly, we assume there is some function that extracts the angles, size, color, or whatever other property of elements we’re interested in. Two elements would be considered equivalent iff the function extracts the same value for each. � Course Notes, Week 4: Binary Relations 3 For example, if fc is the function mapping a triangle to the lengths its sides, then fc determines the congruence relation....

For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphism...