Mathematics & Physics 2016, 9(2), 144–148 УДК 512.53 Bidiagonal Ranks of Completely (0-)simple Semigroups Ilia V. Barkov∗ National Research University of Electronic Technology Shokin square, 1, Moscow, Zelenograd, 124498 Russia Received 12.09.2015, received in revised form 19.01.2016, accepted 24.02.2016 A bidiogonal act over a semigroup is a two-sided act, where the semigroup acts on its Cartesian power. <...> A bidiagonal rank of a semigroup is the least power of a generating set of the bidiagonal act over this semigroup. <...> Keywords: act over a semigroup, diagonal rank, completely (0-)simple semigroup. <...> A generating set G of the act (S ЧS)S is called irreducible if none of its subsets G′ ⊂ G is a generating set of this act. <...> Clearly, any finite generating set may be reduced to an irreducible one. <...> Note that a diagonal act over a semigroup is a unary algebra. <...> Indeed, if S is a semigroup, then multiplication by s ∈ S may be thought as applying unary operation ϕs : x → xs, where x ∈ S. Therefore, the following theorem is applicable to diagonal acts. <...> Let A be an algebra with signature Σ = {ϕi | i ∈ I}, where all operations ϕi are unary. <...> If A is finitely generated, then any irreducible generating set of A is minimal. <...> A right diagonal rank of S (denoted by rdrS) is called the least power of generating sets of the diagonal rigth act of S, or rdrS = min{|A| | A ⊆ S ЧS ∧ AS1 = S ЧS}. <...> For completeness we cite the prime results of [9] concerned with completely (0-)simple ∗zvord@b64.ru ⃝ Siberian Federal University. <...> All rights reserved c – 144 – Ilia V. Barkov Bidiagonal Ranks of Completely (0-)simple Semigroups semigroups. <...> Let S be a Rees matrix semigroup with sandwich-matrix P: S = M(G, I,Λ,P). <...> Then the right diagonal rank of S equals |I|2G, if Λ is singleton and |I|2|G|2Λ(Λ−1) otherwise. <...> Let S be a Rees matrix semigroup with zero: S =M0 (G, I,Λ,P). <...> Let S be a completely <...>