The Reducibility Concept in General Hyperrings

Irina Cristea; Milica Kankaraš

doi:10.3390/math9172037

The Reducibility Concept in General Hyperrings

Mathematics ◽

10.3390/math9172037 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2037

Author(s):

Irina Cristea ◽

Milica Kankaraš

Keyword(s):

Formal Series ◽

Equivalence Relations ◽

General Properties

By using three equivalence relations, we characterize the behaviour of the elements in a hypercompositional structure. With respect to a hyperoperation, some elements play specific roles: their hypercomposition with all the elements of the carrier set gives the same result; they belong to the same hypercomposition of elements; or they have both properties, being essentially indistinguishable. These equivalences were first defined for hypergroups, and here we extend and study them for general hyperrings—that is, structures endowed with two hyperoperations. We first present their general properties, we define the concept of reducibility, and then we focus on particular classes of hyperrings: the hyperrings of formal series, the hyperrings with P-hyperoperations, complete hyperrings, and (H,R)-hyperrings. Our main aim is to find conditions under which these hyperrings are reduced or not.

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Reachability Relations and the Structure of Transitive Digraphs

The Electronic Journal of Combinatorics ◽

10.37236/115 ◽

2009 ◽

Vol 16 (1) ◽

Author(s):

Norbert Seifter ◽

Vladimir I. Trofimov

Keyword(s):

Positive Integer ◽

Negative Integer ◽

Equivalence Relations ◽

Basic Relation ◽

Factor Graphs ◽

Locally Finite ◽

General Properties

In this paper we investigate reachability relations on the vertices of digraphs. If $W$ is a walk in a digraph $D$, then the height of $W$ is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed opposite to their orientation. Two vertices $u,v\in V(D)$ are $R_{a,b}$-related if there exists a walk of height $0$ between $u$ and $v$ such that the height of every subwalk of $W$, starting at $u$, is contained in the interval $[a,b]$, where $a$ ia a non-positive integer or $a=-\infty$ and $b$ is a non-negative integer or $b=\infty$. Of course the relations $R_{a,b}$ are equivalence relations on $V(D)$. Factorising digraphs by $R_{a,\infty}$ and $R_{-\infty,b}$, respectively, we can only obtain a few different digraphs. Depending upon these factor graphs with respect to $R_{-\infty,b}$ and $R_{a,\infty}$ it is possible to define five different "basic relation-properties" for $R_{-\infty,b}$ and $R_{a,\infty}$, respectively. Besides proving general properties of the relations $R_{a,b}$, we investigate the question which of the "basic relation-properties" with respect to $R_{-\infty,b}$ and $R_{a,\infty}$ can occur simultaneously in locally finite connected transitive digraphs. Furthermore we investigate these properties for some particular subclasses of locally finite connected transitive digraphs such as Cayley digraphs, digraphs with one, with two or with infinitely many ends, digraphs containing or not containing certain directed subtrees, and highly arc transitive digraphs.

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