Left Zero Research Articles

Polygons with a (P, 1)-Stable Theory

Polygons with a (P, 1)-stable theory are considered. A criterion of being (P, 1)-stable for a polygon is established. As a consequence of the main criterion we prove that a polygon SS, where S is a group, is (P, 1)-stable if and only if S is a finite group. It is shown that the class of all polygons with monoid S is (P, 1)-stable only if S is a one-element monoid. (P, 1)-stability criteria are presented for polygons over right and left zero monoids.

Domination in Cayley Digraphs of Right and Left Groups

Let Cay(S,A) denote a Cayley digraph of a semigroup S with a connection set A. A semigroup S is said to be a right group if it is isomorphic to the direct product of a group and a right zero semigroup and S is called a left group if it is isomorphic to the direct product of a group and a left zero semigroup. In this paper, we attempt to nd the value or bounds for the domination number of Cayley digraphs of right groups and left groups. Some examples which give sharpness of those bounds are also shown. Moreover, we consider the total domination number and give the necessary and sufficient conditions for the existence of total dominating sets in Cayley digraphs of right groups and left groups.

P-Stable Polygons

P-stable polygons are studied. It is proved that the property of being (P, s)-, (P, a)-, and (P, e)-stable for the class of all polygons over a monoid S is equivalent to S being a group. We describe the structure of (P, s)-, (P, a)-, and (P, e)-stable polygons SA over a countable left zero monoid S and, under the condition that the set A \ SA is indiscernible, over a right zero monoid.

Quantum groups as generalized gauge symmetries in WZNW models. Part I. The classical model

Wess–Zumino–Novikov–Witten (WZNW) models over compact Lie groups G constitute the best studied class of (two dimensional, 2D) rational conformal field theories (RCFTs). A WZNW chiral state space is a finite direct sum of integrable representations of the corresponding affine (current) algebra, and the correlation functions of primary fields are monodromy invariant combinations of left times right sector conformal blocks solving the Knizhnik–Zamolodchikov equation. However, even in this very well understood case of 2D RCFT, the “internal” (gauge) symmetry that governs the ensuing fusion rules remains unclear. On the other hand, the canonical approach to the classical chiral WZNW theory developed by Faddeev, Alekseev, Shatashvili, Gawedzki and Falceto reveals its Poisson–Lie symmetry. After a covariant quantization, the latter gives rise to an associated quantum group symmetry which naturally requires an extension of the state space. This paper contains a review of earlier work on the subject with a special emphasis, in the case G = SU(n), on the emerging chiral “WZNW zero modes” which provide an adequate algebraic description of the internal symmetry structure of the model. Combining further left and right zero modes, one obtains a specific dynamical quantum group, the structure of its Fock representation resembling the axiomatic approach to gauge theories in which a “restricted” quantum group plays the role of a generalized gauge symmetry.

ИНЪЕКТИВНЫЕ И ПРОЕКТИВНЫЕ ПОЛИГОНЫ НАД ВПОЛНЕ 0-ПРОСТОЙ ПОЛУГРУППОЙ

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

Amenability and geometry of semigroups

We study the connection between amenability, Folner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Folner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Folner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Folner condition in terms of the existence of weak Folner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.

On left quasi-ample semigroups with ΠL1-embeddable band of idempotents

ABSTRACTLet ΠL1 denote a direct power of L1, the two-element left zero semigroup with identity adjoined. A semigroup S is called left quasi-ample if for each a∈S there exists a unique idempotent a+∈S such that for all x, y∈S1 and the left ample condition holds. Generalizing a recent result in [3], we prove that the semigroups in the title are embeddable into certain transformation semigroups. Our embedding provides an easy way to construct (finite) proper covers for (finite) such semigroups. Moreover, we show that each proper such semigroup is embeddable into a semidirect product of a ΠL1-embeddable band by a right cancellative monoid, giving a partial answer to a question raised in [1].

Algebras of cubic matrices

We consider algebras of -cubic matrices (with ). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size m. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff . We introduce a notion of accompanying algebra (which is -dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed.

Solvability conditions of problem of synthesizing proportional-integral control of a linear MIMO system

The solvability conditions and just the solution of the problem of the regular and irregular proportional-integral (PI) control are found in accordance with the properties of invariant zeros of a multi-input multioutput (MIMO) system. It is proved that the problem of synthesizing the control of the MIMO system is solvable if and only if the pair of matrices (A, B) that describes a control plant is controllable and the matrix BL?ACR? (where BL? is the left zero divisor of the matrix B and CR? is the right zero divisor of the output matrix C) has a complete row rank.

Infinite compact sets of idempotents in $$\beta S$$ β S

We show that if S is a countably infinite right cancellative semigroup and T is an infinite compact set of idempotents in the Stone–Cech compactification \(\beta S\) of S, then T contains an infinite compact left zero semigroup.

Monoids over which products of indecomposable acts are indecomposable

In this paper we prove that for a monoid $S$, products of indecomposable right $S$-acts are indecomposable if and only if $S$ contains a right zero. Besides, we prove that subacts of indecomposable right $S$-acts are indecomposable if and only if $S$ is left reversible. Ultimately, we prove that the one element right $S$-act $\Theta_S$ is product flat if and only if $S$ contains a left zero.

Isomorphism Conditions for Cayley Graphs of Rectangular Groups

A rectangular band is defined as a direct product of a left zero semigroup and a right zero semigroup, and a rectangular group is defined as a direct product of a group and a rectangular band. In this paper, we give some equivalent conditions for Cayley graphs of a rectangular group to be isomorphic to each other.

On $$\mathcal {R}$$ R -unipotent semigroups with $$\Pi L^1$$ Π L 1 -embeddable band of idempotents

Let $$\Pi L^1$$ denote a direct power of $$L^1$$ , the two-element left zero semigroup with identity adjoined. We prove that the semigroups in the title are embeddable into transformation semigroups which naturally generalize the Vagner one-point completion of the symmetric inverse semigroup. We give some examples and show that our representation provides a convenient way to construct E-unitary covers and embeddings of E-unitary such semigroups into semidirect products of $$\Pi L^1$$ -embeddable bands by groups.

Semigroup graded algebras and codimension growth of graded polynomial identities

We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T-graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T-graded algebra is unital, or T is a cancellative semigroup, then the T-graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.

High gain two‐stage amplifier with positive capacitive feedback compensation

A novel topology for a high gain two-stage amplifier is proposed. The proposed circuit is designed in a way that the non-dominant pole is at output of the first stage. A positive capacitive feedback around the second stage introduces a left half-plane zero, which cancels the phase shift introduced by the non-dominant pole, considerably. The dominant pole is at the output node, which means that increasing the load capacitance has minimal effect on stability. Moreover, a simple and effective method is proposed to enhance slew rate. Simulation shows that slew rate is improved by a factor of 2.44 using the proposed method. The proposed amplifier is designed in a 0.18 µm complementary metal-oxide-semiconductor process. It consumes 0.86 mW power from a 1.8 V power supply and occupies 3038.5 µm2 of chip area. The DC gain is 82.7 dB and gain bandwidth (GBW) is 88.9 MHz when driving a 5 pF capacitive load. Also low frequency common-mode rejection ratio and positive power supply rejection ratio are 127 and 83.2 dB, respectively. They are 24.8 and 24.2 dB at GBW frequency, which are relatively high and are other important properties of the proposed amplifier. Moreover, simulations show convenient performance of the circuit in process corners and also the presence of a mismatch.

A 0.0045- <inline-formula> <tex-math notation="TeX">$\hbox{mm}^{2}$</tex-math></inline-formula> 32.4- <inline-formula> <tex-math notation="TeX">$\mu\hbox{W} $</tex-math></inline-formula> Two-Stage Amplifier for pF-to-nF Load Using CM Frequency Compensation

This brief reports an embedded capacitor multiplier (CM) frequency compensation technique to realize an extremely compact micropower two-stage amplifier for wide capacitive load ( $C_{L} $ ) drivability. It features: 1) a valuable left half-plane zero to enhance the closed-loop stability over a wide range of $C_{L} $ ; 2) no extra bias circuit and power, as the CM is embedded into the first stage of the amplifier, and 3) only one very small (subpicofarad) compensation capacitor improving the transient settling and area efficiency. Detailed analytical treatments of the amplifier offer the critical insights for device sizing and optimization. Fabricated in 0.18- $\mu\hbox{m} $ CMOS, the amplifier measures 3.06-MHz unity-gain frequency (UGF), 1.76- $\hbox{V}/\mu\hbox{s}$ average slew rate (SR), and $74^{\circ}$ phase margin (PM) at 20-pF $C_{L}$ , and 0.22-MHz UGF, 0.049- $\hbox{V}/\mu\hbox{s}$ SR, and $59.8^{\circ}$ PM at 15-nF $C_{L}$ . The die size is 0.0045 $\hbox{mm}^{2}$ , and power is $32.4 \mu\hbox{W} $ at 1.2 V. Competitive large- and small-signal figures of merit are achieved with respect to the state of the art.

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RSn generated by all completely 0-simple semigroups over groups from the Burnside variety Gn of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B2, generated by the five element Brandt semigroup B2, and contained in the variety NB2∨Gn where NB2 is the variety generated by all left and right zero semigroups together with B2. The interval [NB2,NB2∨Gn] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RSn.

Diameter of the zero divisor graph of semiring of matrices over Boolean semiring

Let S be a semiring and let Z(S) be its set of nonzero zero divisors. We denote the zero divisor graph of S by ( S) whose vertex set is Z(S) and there is an edge between the vertices x and y (x6 y) in ( S) if and only if either xy = 0 or yx = 0. In this paper we study the zero divisor graph of the semiring of matrices Mn(B), (n > 1) over the Boolean semiring B. We investigate the properties of the right zero divisors and the left zero divisors of Mn(B) and then use these results to prove that the diameter of ( Mn(B)) is 3.

Semigroups embeddable in hyperplane face monoids

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.

Digital Non-linear Controller-based Active Power Filter for Harmonic Compensation

This paper presents a design and implementation of digital non-linear controller-based, three-phase shunt-active filter for power line conditioner system. The active power filter is implemented with PWM-Voltage Source Inverter and its switching pulses are generated from Space Vector Modulation techniques. The active filter is applied for power quality improvements by compensating harmonics in the distribution network due to non-linear loads. The harmonic components are extracted from the sensed load currents from proposed non-linear control strategy. A simple and efficient Phase Locked Loop is used to synchronize the angle of vector orientation, which cancels the effect of harmonics and notches in the distribution network. An inner and outer loop Pl-control is incorporated with non-linear method for harmonic current control and dc-bus voltage regulator. Additionally, pre-filter is inserted at the input of each current loop in order to compensate for the effect of the left hand zero in the closed loop transfer functions. The entire control method is validated through MATLAB and also coded in assembly language for digital hardware implementation. The digital control algorithm is evaluated through TMS320F240 Digital Signal Processor for 10 kVAR active filter system.