مسابقة دكتوراه 2025 — Centre Universitaire de Barika — الموضوع 02

Let $G$ be a group with identity element $1_G$ and $P$ a subset of $G$ satisfying the following three conditions:

(i) $P \cap P^{-1} = \{1_G\}$, (ii) $P^2 = P$, where and (iii) , .

Consider the binary relation $\leq$ defined on $G$ by: $x \leq y \Leftrightarrow yx^{-1} \in P$.

1. Prove that $(G, \leq)$ is an ordered group.

2. Deduce that $P$ is the positive cone relative to this compatible order on $G$.

3. Prove that $\leq$ is total if and only if $P \cup P^{-1} = G$.

$$(x, y) \in P \Leftrightarrow (x = y = 0) \text{ or } (x \geq 0 \text{ and } y > 0).$$

a) Prove that $P$ is the positive cone of the compatible order $\leq$ on $\mathbb{R} \times \mathbb{R}$.

b) Show that this order is described by:

$$(x, y) \leq (x', y') \Leftrightarrow \begin{cases} x = x' \text{ and } y = y'; \\ \text{or } x \leq x' \text{ and } y < y'. \end{cases}$$

Let $G$ be a group and $H$ be a fuzzy subgroup of $G$. $H$ is called a normal fuzzy subgroup if for all $x, y \in G$, $\mu_H(xy) = \mu_H(yx)$, where $\mu_H(x)$ is the degree of membership of $x$ in the fuzzy set $H$.

1. Prove that $H$ is a fuzzy normal subgroup of $G$ if and only if $\mu_H$ is constant in each conjugate class of $G$.

   Note that: $\mu_H$ is constant in each conjugate class of means that , .

$$\mu_H[x, y] \geq \mu_H(x) \quad \text{for all } x, y \in G,$$

where $[x, y]$ is the commutator of $x$ and $y$, and it is defined by: $[x, y] = x^{-1}y^{-1}xy$.

3. Let $\alpha \in [0, 1]$ such that $\alpha \leq \mu_H(e)$, where $e$ denotes the identity of . Prove that if is a fuzzy normal subgroup of , then is a normal subgroup of .

$$\forall x \in G,\; \mu_{\varphi^{-1}(B)}(x) = \mu_B(\varphi(x)).$$

Prove that $\varphi^{-1}(B)$ is a fuzzy normal subgroup of $G$.