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1

Criticality of Switching Classes of Reversible 2-Structures Labeled by an Abelian Group

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Belkhechine H. , Ille P. , Woodrow R. E.

Discussiones Mathematicae Graph Theory

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2017

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tom 37

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nr 1

175-209

EN

Let V be a finite vertex set and let (𝔸, +) be a finite abelian group. An 𝔸-labeled and reversible 2-structure defined on V is a function g : (V × V) \ {(v, v) : v ∈ V } → 𝔸 such that for distinct u, v ∈ V, g(u, v) = −g(v, u). The set of 𝔸-labeled and reversible 2-structures defined on V is denoted by ℒ(V, 𝔸). Given g ∈ ℒ(V, 𝔸), a subset X of V is a clan of g if for any x, y ∈ X and v ∈ V \ X, g(x, v) = g(y, v). For example, ∅, V and {v} (for v ∈ V) are clans of g, called trivial. An element g of ℒ(V, 𝔸) is primitive if |V | ≥ 3 and all the clans of g are trivial. The set of the functions from V to 𝔸 is denoted by 𝒮(V, 𝔸). Given g ∈ ℒ(V, 𝔸), with each s ∈ 𝒮(V, 𝔸) is associated the switch gs of g by s defined as follows: given distinct x, y ∈ V, gs(x, y) = s(x) + g(x, y) − s(y). The switching class of g is {gs : s ∈ 𝒮(V, 𝔸)}. Given a switching class 𝔖 ⊆ ℒ(V, 𝔸) and X ⊆ V, {g↾(X × X)\{(x,x):x∈X} : g ∈ 𝔖} is a switching class, denoted by 𝔖[X]. Given a switching class 𝔖 ⊆ ℒ(V, 𝔸), a subset X of V is a clan of 𝔖 if X is a clan of some g ∈ 𝔖. For instance, every X ⊆ V such that min(|X|, |V \ X|) ≤ 1 is a clan of 𝔖, called trivial. A switching class 𝔖 ⊆ ℒ(V, 𝔸) is primitive if |V | ≥ 4 and all the clans of 𝔖 are trivial. Given a primitive switching class 𝔖 ⊆ ℒ(V, 𝔸), 𝔖 is critical if for each v ∈ V, 𝔖 − v is not primitive. First, we translate the main results on the primitivity of 𝔸-labeled and reversible 2-structures in terms of switching classes. For instance, we prove the following. For a primitive switching class 𝔖 ⊆ ℒ(V, 𝔸) such that |V | ≥ 8, there exist u, v ∈ V such that u ≠ v and 𝔖[V \ {u, v}] is primitive. Second, we characterize the critical switching classes by using some of the critical digraphs described in [Y. Boudabous and P. Ille, Indecomposability graph and critical vertices of an indecomposable graph, Discrete Math. 309 (2009) 2839–2846].

2

On primitive A-convex algebras

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Tsertos Y.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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tom [Z] 41

203-219

EN

The theory of modules is used in the study of Banach algebras to get, for instance, results of Le Page, on commutativity, Jacobson radical, semicimplicity, algebraic density and the like (cf., for example, [1], [2]). Such type of results were obtained in [8] in the context of lmc Q-algebras, while in the present paper there are considered, more generally, for A-convex Q'-algebras, whose class is proved thus to be a convenient setting thereof.

3

On primitive A-convex algebras

100%

Tsertos Y.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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tom [Z] 41

203-219

EN

The theory of modules is used in the study of Banach algebras to get, for instance, results of Le Page, on commutativity, Jacobson radical, semicimplicity, algebraic density and the like (cf., for example, [1], [2]). Such type of results were obtained in [8] in the context of lmc Q-algebras, while in the present paper there are considered, more generally, for A-convex Q'-algebras, whose class is proved thus to be a convenient setting thereof.