Mathematics & Physics 2016, 9(2), 220–224 УДК 519.7 Minimal Algebras of Unary Multioperations Nikolay A.Peryazev∗ Saint Petersburg Electrotechnical University Professor Popov, 5, Saint Peterburg, 197376 Russia Yulia V.Peryazeva† Gymnasium 24 of Saint Petersburg Srednii Avenue, 20, Saint Peterburg, 199053 Russia Ivan K. Sharankhaev‡ Institute of Mathematics and Computer Science Buryat State University Smolin, 24a, Ulan-Ude, 670000 Russia Received 10.01.2016, received in revised form 17.02.2016, accepted 24.03.2016 A matrix impression of algebras of unary multioperations of a finite rank and the list of the identities which are carried out in such algebras are gained. <...> These results are used for the proof of the main result: descriptions of the minimal algebras of unary multioperations of a finite rank. <...> Introduction Algebras of unary multioperations which are considered in this paper are finite algebras. <...> The main result of this paper was announced in [3]. <...> There is the following matrix representation of algebras of unary multioperations. <...> Boolean matrices are binary matrices on the elements which define the Boolean operations. <...> For unary multioperation f on A we define Boolean square matrix Mf = (αij) of order k as follows: αij = 1 if ai ∈ f(aj) else αij = 0. <...> Operations of algebra of unary of multioperations are represented by matrix operations in the following way: Mf∗g =Mf ∗Mg is matrix multiplication; Mf∩g =Mf ◦Mg is element-wise matrix multiplication; Mµf =MT f is transposition of matrix; Mε = E is diagonal matrix; Mθ = O is null matrix; Mπ = P is unit matrix. <...> The main result The smallest algebra which not equal trivial algebra consisting of only multioperations π, θ, ε is called minimal algebra of unary multioperations. <...> It is obvious that necessary and sufficient condition for minimality of algebra of unary multioperations is the generating of any its multioperation which not equal π, θ, ε. <...> The following theorem describes the multioperations generating minimal algebras of unary multioperations. <...> In addition each multioperation other than π, θ, ε generates all elements of its algebra. <...> We now show <...>