Abstract
Our main purpose in this work is to study the maximal chains in group-posets to observe that this study gives us indications on the type of some group actions on posets. Therefore we shall study the behavior of the group actions on chains .
Highlights
For any group G and any set X, we say that G acts on X from the left if to each g∈G and x∈X there corresponds a unique element in X denoted by g x such that for all x∈X and g1,g2∈G;(i) e x = x (ii) g1(g2 x) =(g1g2 ) x .Such a set X with a left action of G on it, is called a left G-set, or a G-set. [13].Since the concept of a group action of a group G on a set X began as a group homomorphism ρ : G →S1x1, we can consider any element g in G as a permutation g : X →X with g(x) =g x for all x∈X
This concept can be extended on sets with additional mathematical structure, with ρ : G →isom (X,X) and the isomorphism related to the structure on X
Group-posets : we give the definition of the group actions on posets
Summary
For any group G and poset P there is at least the trivial action which defined by : g p = p for all g∈G , p∈P. For each p∈P , the set {g∈G: g p = p} is called the stabilizer of p and denoted by StabG(p) or Gp. Proposition (2-5) : [8] Let P be a G-poset. Proposition (2-6) : Let P be a G-poset. Proposition (2-8) : Let P be a G-poset and a,b ∈P with a covers b , g a covers g b for all g∈G.
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