Pair Of Idempotents Research Articles

A flow of quantum genetic Lotka-Volterra algebras on [formula omitted

In the present paper, we define a quantum analog of Lotka-Volterra algebras using a coalgebra scheme. This new framework provides a fresh perspective for the treatment of generic algebras. Additionally, a flow of quantum analogs of Lotka-Volterra genetic algebras is investigated. It's worth mentioning that such types of algebras are first introduced in this work. We observe that a flow of algebras is a particular type of continuous-time dynamical system, with states that are algebras and a structural constant matrix that depends on time and satisfies certain analogs of the Kolmogorov-Chapman equations. Using quantum quadratic operators, it is constructed a flow of quantum Lotka-Volterra algebras for the given multiplication. Furthermore, time-dependent behavior properties of these flow algebras are examined. The algebraic properties of the introduced flows are also studied, such as finding idempotents and examining an algebra generated by a pair of idempotents. It is shown that the later one is associative, while the flow is not associative. Additionally, derivations of the flow of the algebras are described.

Invariant Subspaces of Idempotents on Hilbert Spaces

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.

Skew Pairs of Idempotents in Partial Transformation Semigroups

Let S be a regular semigroup. A pair (e,f) of idempotents of S is said to be a skew pair of idempotents if fe is idempotent, but ef is not. T. S. Blyth and M. H. Almeida (T. S. Blyth and M. H. Almeida, skew pair of idempotents in transformation semigroups, Acta Math. Sin. (English Series), 22 (2006), 1705–1714) gave a characterization of four types of skew pairs—those that are strong, left regular, right regular, and discrete—existing in a full transformation semigroup T(X). In this paper, we do in this line for partial transformation semigroups.

Jordan-Lie inner ideals of finite dimensional associative algebras

We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

Invariant and Almost-Invariant Subspaces for Pairs of Idempotents

It is known that every bounded operator on an infinite dimensional separable Hilbert space \({\mathcal{H}}\) has an invariant subspace if and only if each pair of idempotents on \({\mathcal{H}}\) has a common invariant subspace. We show that the same equivalence holds for operators and pairs of idempotents that are essentially selfadjoint. We also show that each pair of idempotents on \({\mathcal{H}}\) has a common almost-invariant half-space.

Skew idempotents in linear transformation semigroups

If S is a regular semigroup, then (e, f) is a skew pair of idempotents in S if e, f and fe are idempotent, but ef is not. In 2006, Blyth and Almeida Santos determined when four types of such skew pairs exist in T(X), the total transformation semigroup on a set X. Here we do the same for the semigroup T(V) consisting of all linear transformations of a vector space V.

Derived Rees Matrix Semigroups as Semigroups of Transformations

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that TX contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that TX contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.

On the diameters of commuting graphs

The commuting graph of a ring R , denoted by Γ ( R ) , is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ⩾ 3. In this paper we investigate the diameters of Γ( M n ( D)) and determine the diameters of some induced subgraphs of Γ( M n ( D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M n ( D). For every field F, it is shown that if Γ( M n ( F)) is a connected graph, then diam Γ( M n ( F)) ⩽ 6. We conjecture that if Γ( M n ( F)) is a connected graph, then diam Γ( M n ( F)) ⩽ 5. We show that if F is an algebraically closed field or n is a prime number and Γ( M n ( F)) is a connected graph, then diam Γ( M n ( F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.

Skew Pairs of Idempotents in Transformation Semigroups

An ordered pair (e, f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we investigate the ubiquity of such skew pairs in full transformation semigroups.

The Smallest Regular Semigroups with all Four Skew Pairs of Idempotents

An ordered pair (e, f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we consider the smallest regular semigroups that contain precisely one of each of these four types. We show that, to within isomorphism and dualisomorphism, there are six such semigroups and characterise them as quotient semigroups of certain regular Rees matrix semigroups.

A characterization of commutators of idempotents

Necessary and sufficient conditions are obtained for an element in a ring to be a commutator of idempotents. In the special case of finite-dimensional operators, it is shown that an operator T is a commutator of a pair of idempotents if and only if T is similar to − T and the Riesz part T 1 of T corresponding to 1 2 i has the property that T 1 2+ 1 4 I has a square root.

On Commutators of Idempotents

It is shown that a pair of idempotent operators on a Banach space is triangularizable if their commutator is nilpotent. Moreover, if every operator on Hilbert space has an invariant subspace, then a pair of idempotents on Hilbert space is triangularizable if their commutator is quasinilpotent. These results are generalized from idempotents to quadratic operators.

On the complexity of description of representations of $*$-algebras generated by idempotents

In this paper, we introduce a quasiorder $\succ$ (majorization) on $*$-algebras with respect to the complexity of description of their representations. We show that $C^*({\mathcal F}_2) \succ \mathfrak A$ for any finitely generated $*$-algebra $\mathfrak A$ (algebras $\mathfrak B$ such that $\mathfrak B\succ C^*({\mathcal F}_2)$ are called $*$-wild). We show that the $*$-algebra generated by orthogonal projections $p$, $p_1$, $p_2$, …, $p_n$ ($p_ip_j=0$ for $i\neq j$) is $*$-wild if $n\geq 2$. We also prove that $*$-algebras generated by a pair of idempotents and an orthogonal projection, or by a pair of idempotents $q_1$, $q_2$ ($q_1q_2=q_2 q_1=0$), etc., are $*$-wild.

A geometric equivalent of the invariant subspace problem

It is shown that every operator has an invariant subspace if and only if every pair of idempotents has a common invariant subspace.

A note on representations of inverse semigroups

It is known [1; 2] that every inverse semigroup S has a faithful representation as a semigroup of (1, 1)-mappings of subsets of a set A into A. The set A may be taken as the set of elements of S and the (1, 1)-mappings as mappings of principal left ideals of S onto principal left ideals of S. If E is the set of idempotents of S then there is also a representation of S, not necessarily faithful, as a semigroup of (1, 1)-mappings of subsets of E into E [2]. If eEE denote by S. the subsemigroup eSe of S. In this note we give a representation of any inverse semigroup S as a semigroup of isomorphisms between the semigroups Se, The representation is faithful if (a more general condition is given below) the center of each maximal subgroup of S is trivial. We recall that an inverse semigroup [3] is a semigroup S in which for any a E S the equations xax = x and axa = a have a unique common solution xES called the inverse of a and denoted by a-' [5; 6]. This implies that the idempotents of S commute and that to each aGS there corresponds a pair of idempotents e, f such that aa= e, a-la =f, ea = a, af = a. The idempotents e, f are called respectively the left and right units of a. For any two elements a, bES, (ab)-'=b-1a(see [3]). Throughout what follows S will denote an inverse semigroup and E will denote its set of idempotents. If eEE then Se will denote the subsemigroup eSe of S.