Fuzzy linear equations systems.

Necessary familiar concepts from fuzzy numbers theory and interval systems are presented in the article, namely, concepts of: a fuzzy number; the carrier and the membershi function of fuzzy number; a discrete and continual fuzzy number; an interval matrix; low and upper bounds of an interval matrix; a middle matrix; a matrix of radiuses; an interval vector; low and upper bounds of an interval vector; a middle vector; the vector of radiuses.
New concepts such as concepts of a one-peak fuzzy number (discrete and continual); an acute and not acute peak; a normal fuzzy number; a standard fuzzy number; a standardized fuzzy number are introduced in the work.
The concepts of a fuzzy matrix, which connects interval apparatus and fuzzy numbers apparatus; interval linear systems of equations; the fuzzy linear system of equations are introduced in the work.
The concept of admitable with the type of membership $< t, τ >$ solution of an uncertain linear system of equations is introduced in the work. Characterization of a solution is given.
It is proved the lemma that if $I^t_A$ is an interval matrix, presented through its low $\underline{A}^t$ and upper $\bar{A}^t$ bounds $[\underline{A}^t,\bar{A}^t]$, where $\bar{A}^t, \bar{A}^t \in R^{m×n}$, and $x$ is admitable with the type of membership $< t, τ >$ solution, $x \in R^n$, then the family with the number $t$ of right parts of systems of linear equations $A^tx$ can be presented through the matrix of radiuses $Δ^t: \{A^tx | A^t \in I^t_A\} = [A^t_cx − Δ^t|x|, A^t_cx + Δ^t|x|]$.
It is proved for the family with the number $t A^tx$ of right parts of systems of linear equations, that next statements are equivalent: 1) $x$ is admitable with the type of membership $< t, τ >$ solution of uncertain linear system of equations of the form $F_Ax = F_b$; 2) $x$ satisfies the inequality $|A^t_cx − b^τ_c| ≤ −Δ^t|x|+δ^τ$, where $A^t_c$ is the middle matrix of the interval matrix, $b^τ_c$ is the middle vector of the interval vector, $Δ^t$ is the matrix of radiuses of the interval matrix, $δ^τ$ is the vector of radiuses of the interval vector.
3) $x = x_1 − x_2$, where $x_1, x_2$ satisfy conditions:$\bar{A}^tx_1 − \underline{A}^tx_2 ≤ \bar{b}^τ; \underline{A}^tx_1 − \bar{A}^tx_2 ≥ \underline{b}^τ; x_1 ≥ 0; x_2 ≥ 0$, where $\underline{A}^t, \bar{A}^t$ are low and upper bounds of the interval matrix; $\underline{b}^τ,\bar{b}^τ$ are low and upper bounds of the interval vector.
It is substantiated that the check that x is admitable with the type of membership $< t, τ >$ solution of the uncertain system can be executed for the polynomial time.