Algebraic extension of $\mathcal{A}^{*}_{n}$ operator

$T\in L(H_{1}\oplus H_{2})$ is said to be an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator if $$ T = \begin{pmatrix} T_{1} & T_{2} \\O & T_{3} \end{pmatrix} $$ is an operator matrix on $H_{1}\oplus H_{2}$, where $T_{1}$ is a $\mathcal{A}^{*}_{n}$ operator and $T_{3}$ is a algebraic.In this paper, we study basic and spectral properties of an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator. We show that every algebraic extension of a $\mathcal{A}^{*}_{n}$ operator has SVEP, is polaroid and satisfies Weyl's theorem.

On Pairs of Additive Subgroups Associated with Intermediate Subgroups of Groups of Lie Type over Nonperfect Fields

The author has previously (Trudy IMM UrO RAN, 19(2013), no. 3) described the groups lying between twisted Chevalley groups G(K) and G(F) of type 2Al, 2Dl, 2E6, 3D4 in the case when the larger field F is an algebraic extension of the smaller nonperfect field K of exceptional characteristic for the group G(F) (characteristics 2 and 3 for the type 3D4 and only 2 for other types). It turned out that apart from, perhaps, the type 2Dl, such intermediate subgroups are standard, that is, they are exhausted by the groups G(P)H for some intermediate subfield P, K ⊆ P ⊆ F, and of the diagonal subgroup H normalizing the group G(P). In this note, it is established that intermediate subgroups are also standard for the type 2Dl

On EMV-Semirings

The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.