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For n ∈ Z + , let X n = { 1 , ... , n } be a finite chain and T n be the full transformation semigroup on Xn. If O C T n is the set of all order-preserving and full contraction mappings on Xn, then it is well-known that O C T n is a submonoid of T n . In this paper we give a monoid presentation of O C T n . [ABSTRACT FROM AUTHOR]

In this paper, we study a class of skew cyclic codes over the ring $ R = \mathbb{Z}_4+u\mathbb{Z}_4 $, with $ u^2 = 1 $. We determine a complete structure of skew cyclic codes over $ R $ by investigating the generating sets of these codes. Additionally, we have identified some more conditions on generating polynomials for these codes when dealing with odd lengths, and in this case, the skew-cyclic codes are equivalent to cyclic codes. Some examples have been given to illustrate the results. Moreover, we present some examples of skew cyclic codes over $ R $ whose $ \mathbb{Z}_4 $ images give some new linear codes over $ \mathbb{Z}_4 $. [ABSTRACT FROM AUTHOR]

Let V be a vector space and U a fixed subspace of V. We denote the semigroup of all linear transformations on V under composition of functions by L(V). In this paper, we study the semigroup of all linear transformations on V whose restrictions belong to the general linear group GL(U), denoted by L GL (U) (V) . More precisely, we consider the subsemigroup L GL (U) (V) = { α ∈ L (V) : α | U ∈ GL (U) } of L(V). In this work, Green's relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when V is a finite dimensional vector space over a finite field. Finally, we find its generating set. [ABSTRACT FROM AUTHOR]

Let S be a subset of a finite group Γ. The Cayley signed graph, denoted by Σ = CayΣ(S, σ) has vertex set Γ and two distinct vertices x, y ∈ Γ are joined by an edge from x to y if and only if there exists s ∈ S such that x = sy, where Σ is a signed graph whose underlying graph is Γ and σ : E(Γ) → {+, -} is a function defined as ... In this manuscript, we have characterized the Cayley set and generating sets for which Cayley signed graphs are canonically consistent. [ABSTRACT FROM AUTHOR]

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner–Preston Theorem. Let n be a positive natural number and PFI n be the semigroup of all fence-preserving partial one-to-one maps of { 1 , ... , n } into itself with respect to composition of maps and the fence 1 ≺ 2 ≻ 3 ≺ ⋯   n. There is considered the inverse semigroup IOF n par of all α ∈ PFI n such that α is regular in PFI n , order-preserving with respect to the order 1 < 2 < ⋯ < n and parity-preserving. According to the main result of the paper, it is 3 n − 6 the least of the cardinalities of the generating sets of IOF n par for 4 ≤ n. There is determined a concrete representation of a generating set of minimal size. [ABSTRACT FROM AUTHOR]

In this paper, we study generating sets of the complete semigroup of binary relations defined by X"=semilattices of unions of the class Σ1(X, 6). [ABSTRACT FROM AUTHOR]

The article deals with the issue of ensuring the sustainable operation of ship electrical power systems. Experimental oscillograms of power exchange and in-phase oscillations during parallel operation of diesel generator units are presented. The results of mathematical modeling allow us to conclude that it is necessary to limit the change in the values and ratios of the transmission coefficients of the governor. It is proposed to limit the change in the transmission coefficients of the governor when eliminating power exchange oscillations.

Unmanned seagoing vessels are increasingly being used. In the paper the dump scow Rabochaya is considered as an example of an unmanned vessel. An extended marine control system, a video surveillance subsystem, a surveillance and search engine are additionally installed on the scow to carry out industrial tasks in unmanned mode. The correct operation of these systems and the whole vessel depends on the high-quality operation of the ship's electrical power system. The aim of the study is to identify abnormal modes of parallel operation of marine diesel generator sets and to develop methods and means for their elimination. The results of the experimental research and mathematical modeling have revealed the existence of power exchange oscillations when the ship generators operating in parallel. Power exchange oscillations can be detected with the help of generator current oscillations. It has been proposed to introduce a unit for eliminating power exchange oscillations (EPEO) into the automatic control system of ship electrical power systems for eliminating such oscillations. An algorithm for the operation of the EPEO unit has been developed, it controls the settings of the rotational speed regulators of diesel-generator sets. The proposed methods and tools reduce the amplitude of power exchange oscillations to an acceptable value, and therefore contribute to the high-quality operation of the ship's electrical power system and control systems in unmanned mode. The results obtained make it possible to ensure reliable and trouble-free operation of any autonomous electrical power system during the operation mode without maintenance personnel.

For any two non-empty (disjoint) chains X and Y, and for a fixed order-preserving transformation θ: Y → X, let GO(X, Y; θ) be the generalized order-preserving transformation semigroup. Let O(Z) be the orderpreserving transformation semigroup on the set Z = X∪Y with a defined order. In this paper, we show that GO(X, Y; θ) can be embedded in O(Z, Y) = {α ∈ O(Z): Zα ⊆ Y}, the semigroup of order-preserving transformations with restricted range. If θ ∈ GO(Y,X) is one-to-one, then we show that GO(X, Y; θ) and O(X, im(θ)) are isomorphic semigroups. If we suppose that |X| = m, |Y| = n, and |im(θ)| = r where m, n, r ∈ N, then we find the rank of GO(X, Y; θ) when θ is one-to-one but not onto. Moreover, we find lower bounds for rank(GO(X, Y; θ)) when θ is neither one-to-one nor onto and when θ is onto but not one-to-one. [ABSTRACT FROM AUTHOR]

The development of AC distribution systems provides for the seamless integration of low-voltage microgrids with distributed energy resources (DERs). This poses new challenges for the control of normal, emergency, and post-emergency states of microgrids, calling for the creation and development of information and communications technology infrastructure. Power converters/inverters that are used to integrate renewable DERs lack inertia. Along with them, fossil fuel-fired generation units are also being integrated into microgrids. These include gas generator sets, diesel generator sets, and microturbines, having small (up to 1–2 s) values of mechanical inertia constants—Tj. This leads to an increase in the rate of transients by a factor of 5–10. Under these conditions, the technical requirements for the speed of automatic power flow control systems, as well as the methods they rely on, have to be reconsidered. Microgrids include DC microgrids, AC microgrids, and hybrid (AC-DC) microgrids. In the case of hybrid microgrids, DERs are connected to the DC grid and are integrated into the AC grid through a common inverter. The complexity of the task of microgrid control is due to the need to choose properly the type and extent of control actions so as to prevent the emergence and development of accidents. The employed control methods must ensure the reliable power supply to consumers and the quality of power in microgrids, as well as the reliable operation of the external distribution systems into which they are integrated. The article gives an overview of control methods for low-voltage AC and AC-DC microgrids, which allow one to tackle effectively solve the tasks. [ABSTRACT FROM AUTHOR]

THE PURPOSE. Consider the issue of the presence of distortions in the form of sinusoidal voltage and current in autonomous electrical power systems with electrical propulsion systems, built on the principle of the unified electric power system. Compare the results of the study of voltage distortion for electrical power systems with electrical propulsion systems of various structures and give recommendations for their application.METHODS. For research, unified electric power systems with DC electrical propulsion systems on the Yeysk ferry and alternating current on the asymmetric icebreaker Baltika are considered. The possibilities of programming with frequency control of modern drives of rudder propellers with AC motors are analyzed.RESULTS. Oscillograms of voltages and currents of generators were obtained using thyristor converters and inverters to control electrical propulsion systems in various modes. Significant ripples and distortions of the sinusoidal voltage and generator currents were noted when thyristor converters are used to power DC propulsion electric motors. Also, small deviations from the sinusoidal form of the ship's network voltage were recorded when inverters were used to control AC propulsion motors as part of rudder propellers.CONCLUSION. The issue of ensuring the quality of electricity produced in autonomous electrical power systems of sea vessels is of great relevance and importance. To ensure the best results, it is advisable to use azimuthing podded drive with AC electric motors, the rotation speed of which is implemented by inverter frequency converters with a DC link.

THE PURPOSE. Consider the use of special purpose vessels with a dynamic positioning system. To conduct observations and research of the operation of electrical power systems of such vessels for the existence of problems in them that affect the performance of technological operations. To propose methods and means aimed at eliminating situations and processes that adversely affect the operation of shipboard electrical power systems when laying pipelines.METHODS. For observation and research, the unified electrical power systems with electric rudder propellers and a system for laying an underwater pipeline of the "Seven Vega" cable layer was considered. Collected information on all the main electrical systems of the vessel "Seven Vega". Experimental studies were carried out aimed at studying the operating modes of the electrical power system during the laying of pipelines.RESULTS. Experimental oscillograms of currents of parallel operating diesel-generator sets in different modes have been obtained. The existence of power exchange oscillations during the operation of the unified electric power system of the cable-laying ship “Seven Vega” is noted. The data on the negative influence of power oscillations on the operation of tensioners during the laying of underwater pipelines were obtained.CONCLUSION. The use of sea-going vessels with dynamic positioning for laying pipelines and performing oil and gas production operations requires ensuring the operation of their electrical power systems. To ensure the high-quality operation of tensioners in pipeline laying systems, it is necessary to ensure the adjustment of their regulators and to exclude the occurrence of power exchange oscillations during parallel operation of generator sets.

THE PURPOSE. Consider the use of propeller electric installations as part of ship electrical complexes with a single electric power system. Highlight the rudder drives as a special type of electric propulsion of ships in northern latitudes. Investigate unified electric power systems with a propeller electric installation for the existence of power exchange oscillations in them. Propose methods and means for eliminating power oscillations in such systems.METHODS. To carry out the research, a single electric power system with electric rudder propellers of the world's only asymmetric icebreaker Baltika was considered. All the main elements of such system have been analyzed in detail. Experimental studies were carried out aimed at studying the operating modes of a unified electric power system.RESULTS. Experimental oscillograms of currents of parallel operating diesel-generator sets in different modes have been obtained. The existence of exchange and in-phase power oscillations during the operation of the unified electric power system of the icebreaker "Baltika" is noted. The data on the negative influence of power oscillations on the operation of the electrical complex of the icebreaker are presented.CONCLUSION. The use of ice-class sea vessels is an extremely important task for the Russian Federation. The installation of blocks that eliminate exchange and in-phase power oscillations will improve the reliability and efficiency of the use of marine vessels with electric rudder propellers when servicing hydrocarbon production on the Arctic shelf.

Let X be an infinite set and let Y be a non-empty subset of X. Then T (X, Y) is the semigroup of all full transformations from set X to set Y which is called the transformation semigroup with restricted range. In this paper, we determine the relative rank T (X, Y) modulo the top J class. Moreover, we give examples of the relative rank T (X, Y) modulo S(X, Y) and modulo E(X, Y) to obtain that the relative rank is infinite for a finite set Y. [ABSTRACT FROM AUTHOR]

Let X and Y be two non-empty (disjoint) sets, and let GT (X , Y) denote the set of all (full) transformations from X to Y. For any fixed transformation θ : Y → X , define a sandwich operation ∗ on GT (X , Y) by α ∗ β = α ∘ θ ∘ β for all α , β ∈ GT (X , Y) . Then, the set GT (X , Y) together with the sandwich operation ∗ is a semigroup which is called a generalized transformation semigroup, and it is denoted by GT (X , Y ; θ) . Let T (Z) denote the full transformation semigroup on the set Z = X ∪ Y . In this paper, first we show that GT (X , Y ; θ) can be embedded in T (Z , Y) = { α ∈ T (Z) : Z α ⊆ Y } , the semigroup of transformations with restricted range, which is a subsemigroup of T (Z) . If θ : Y → X is one to one, then we show that GT (X , Y ; θ) and T (X , im (θ)) are isomorphic semigroups. Moreover, if X = m , Y = n , and im (θ) = r where m , n , r ∈ N , then we show that rank (GT (X , Y ; θ)) = ∑ k = r + 1 min (m , n) n ! S (m , k) (n - k) ! if θ is neither onto nor one to one, i.e. r < m and r < n , n m if θ is onto but not one to one, i.e. r = m < n , S (m , n) if θ is one to one but not onto, i.e. r = n < m , where S(m, n) is the Stirling number of the second kind. [ABSTRACT FROM AUTHOR]

In this paper we consider variants of the power set and the lattice of subspaces and study automorphism groups of these variants. We obtain irreducible generating sets for variants of subsets of a finite set lattice and subspaces of a finite vector space lattice. We prove that automorphism group of the variant of subsets of a finite set lattice is a wreath product of two symmetric permutation groups such as first of this groups acts on subsets. The automorphism group of the variant of the subspace of a finite vector space lattice is a natural generalization of the wreath product. The first multiplier of this generalized wreath product is the automorphism group of subspaces lattice and the second is defined by the certain set of symmetric groups.

THE PURPOSE. Consider the most pressing and important problems of parallel operation of generator sets as part of autonomous electrical complexes of modern sea vessels, such as the transition of diesel generator sets to reverse power mode and the existence of power exchange oscillations during parallel operation. Consider the possibility of improving the method for eliminating power exchange oscillations using maps of dependences of the amplitude of oscillations on the parameters of diesel generator sets. METHODS. When solving the problem, a mathematical model of an autonomous electrical complex was used, which allows to take into account the backlash clearances in the control loops of the rotation frequency of parallel diesel generator sets. RESULTS. A map of the dependence of the load angle on the backlash clearances was constructed. The complete identity of the maps of the dependence of the load angle on the backlash clearances and the maps of the dependence of the amplitude of the power exchange oscillations on the backlash gaps has been proved. In the criterion for determining the value of the amplitude of the power exchange oscillations, it is proposed to use the load angle of the generators instead of the envelope currents of the generators. CONCLUSION. The use of the load angle of parallel operating generators allows to increase the speed and accuracy of the unit for eliminating power exchange oscillations. The developed approach makes it possible to control the transition of generating sets to the reverse power mode and the appearance of power exchange oscillations based on measuring the load angles of the generators.

Container ships represent the fastest growing segment of global shipping. Tackling such challenges as the constant pursuit of reduction of unit transportation costs and negative effects on the environment has led to the development of very- and ultra-large container ships, which dominate transport on the world’s major shipping routes. Very-large container ships are conventionally assumed to have a cargo capacity above 10,000 TEU and an overall hull length below 369 meters, whilst the unique feature of ultra-large container ships is an overall hull length over 390 meters. The ship operational states and the main electric power receivers such as reefers and bow thrusters are mentioned and formulae to predict their electricity demand are given. Contemporary configurations of ships’ electric power stations are presented and discussed. Cargo capacity expressed in 20-foot equivalent units (TEU) was identified as the main predictor of the electricity generation capacity based on a representative very- and ultra-large container ships database. Using a simple linear regression model based on the least squares method, a formula was developed to predict the electricity generation capacity of very- and ultra-large container ships at the initial design stage. The prediction will find application in the design of very and ultra-large container ships’ electric power stations. The use of the developed formula at the initial design stage enables obtaining results that are sufficiently accurate to the results of the exact verification calculations at the technical design stage. [ABSTRACT FROM AUTHOR]

For n ∈ N , let OPE n , k be the monoid of all orientation-preserving k-extensive full transformations on { 1 , ... , n } ordered in the standard way. In this paper, it is shown that OPE n , k is an abundant semi-band, and a sufficient and necessary condition of an element in OPE n , k to be regular is obtained. Moreover, a minimal generating set and the minimum idempotent generating set of the monoid OPE n , k are determined, and so the rank and the idempotent rank of OPE n , k are established. [ABSTRACT FROM AUTHOR]

Let { G i : i ∈ N } be a family of finite Abelian groups. We say that a subgroup G ≤ ∏ i ∈ N G i is order controllable if for every i ∈ N , there is n i ∈ N such that for each c ∈ G , there exists c 1 ∈ G satisfying c 1 | [ 1 , i ] = c | [ 1 , i ] , s u p p (c 1) ⊆ [ 1 , n i ] , and order (c 1) divides order (c | [ 1 , n i ]) . In this paper, we investigate the structure of order-controllable group codes. It is proved that if G is an order controllable, shift invariant, group code over a finite abelian group H, then G possesses a finite canonical generating set. Furthermore, our construction also yields that G is algebraically conjugate to a full group shift. [ABSTRACT FROM AUTHOR]

The application of autonomous electrical complexes with parallel diesel generator sets based on synchronous generators is considered. The problem of the existence of exchange power fluctuations in such technical systems is described. A mathematical model has been developed that allows conducting research of an autonomous electrical complex. It is proposed to implement graphical visualization of the dependence of the amplitude of power exchange fluctuations on time constants in the speed control circuits of diesel generators in the form of configuration maps. An example of using the proposed method for a specific electrical complex is given. The possibility of reducing the amplitude of exchange power fluctuations by reducing the values of time constants of frequency control circuits is justified, which is the basis of a new method for eliminating exchange power fluctuations and makes it possible to improve the method, which is based on changing the transmission coefficients of frequency regulators.

Supposing G is a group and k a natural number, dk(G) is defined to be the minimal number of elements of G of order k which generate G (setting dk(G) = 0 if G has no such generating sets). This paper investigates dk(G) when G is a finite Coxeter group either of type Bn or Dn, or of exceptional type. Together with the work of Garzoni and Yu, this determines dk(G) for all finite irreducible Coxeter groups G when 2 ≤ k ≤ rank(G) (rank(G)+1 when G is of type An ). [ABSTRACT FROM AUTHOR]

The problem of the existence of common-mode and exchange power fluctuations when parallel operation of synchronous generators in autonomous electrical systems is considered. The methods of mathematical modeling obtained the amplitude and period of common-mode oscillations in the whole possible range of settings of the object under study. A method has been developed for constructing maps of the dependence of the amplitude and period of common-mode power oscillations on the values of the transfer coefficients and setpoints in terms of the rotation frequency of the speed governors of diesel generators. The use of such cards allows you to apply the method of eliminating exchange power fluctuations, taking into account the possible occurrence of common-mode oscillations. Mapping the settings of the amplitude and the period of the common-mode power oscillations makes it possible to understand the tendency of the amplitude and the period of the common-mode power oscillations to change and to carry out an in-depth analysis of the operation of the autonomous electrotechnical complex.

Let Cn be the semigroup of all order-preserving and decreasing transformations on X = {1, . . ., n} under its natural order, and let N(Cn) be the subsemigroup of all nilpotent elements of Cn . For 1 ≤ r ≤ n − 1, let N(Cn,r) = {α ∈ N(Cn) : |im (α)| ≤ r}, Nr(Cn) = {α ∈ N(Cn) : α is an m-potent for any 1 ≤ m ≤ r}. In this paper we find the cardinality and the rank of the subsemigroup N(Cn,r) of Cn . Moreover, we show that the set Nr(Cn) is a subsemigroup of N(Cn) and then, we find a lower bound for the rank of Nr(Cn). [ABSTRACT FROM AUTHOR]

In 1980 White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our result (Lasoń and Michałek in Adv Math 259:1–12, 2014) confirming White's conjecture 'up to saturation'. Furthermore, we study degrees of Gröbner bases and Betti tables of the toric ideals of matroids of a fixed rank. [ABSTRACT FROM AUTHOR]

The application of autonomous electric power equipment having in their composition parallel operating diesel-generator sets on the basis of synchronous generators is considered. The problem of existence of exchange power fluctuations in such technical systems is described. A mathematical model has been developed that makes it possible to carry out investigations of an autonomous electric power equipment. The technique for constructing maps of the amplitude of the exchange power fluctuations, from the backlash gaps in the speed control circuits of diesel generators is obtained. The use of such cards makes it possible to introduce recommendations on limiting the amplitude of exchange power fluctuations to normative documents, both of a general nature, and specifically for certain types of equipment. In the same way, the cards make it possible to determine the gap gaps existing in the frequency control loop from the measured amplitudes of the exchange power fluctuations.

In low-to-medium power generating sets, a self-powered brushless excitation system is typically employed. This solution is cost-effective, simple and compact, but it suffers from an unreliable voltage build-up at start-up, a slow dynamic response and a relatively low efficiency for the exciter. The push towards more effective, reliable and efficient products has recently led to consider excitation systems equipped with permanent magnet exciters and controlled rotating converters, but their diffusion is limited by their higher complexity and cost. This article investigates the utilization of a hybrid excitation for the exciter, aiming to join the benefits of field windings and permanent magnets. As a case study, this concept is applied to a commercial mid-size generating set adopting an industrial perspective, aiming to maximize the benefits while minimizing the required modifications in the system design. After a preliminary analysis, a consequent-pole layout with surface-mounted bonded magnets is then selected as the most effective solution. Theoretical considerations, numerical analysis and experimental validation are reported to show that the hybrid excitation concept can actually lead to a significant reduction of the exciter field losses as well as to other appreciable side benefits with a very limited impact on the present design of the generating set. [ABSTRACT FROM AUTHOR]

In chemistry, point groups are used to describe the molecular symmetries of a molecule. Meanwhile, a crystallographic point group is a set of point group that holds at least one point in a fixed position and have some restriction on the rotational symmetries. In this research, the application of graph theory and group theory on the symmetry study of molecules is presented in which the Cayley graph of crystallographic point groups under the hexagonal crystal system is determined. The generators are also computed for groups C3h, C6, C6h, D3h, C6v, D6and D6h then used to determine the Cayley graph, which is a directed graph representing the group in terms of the generating sets. This research shows that isomorphic crystallographic point groups produce a similar structure to the Cayley graph but different in labelling. [ABSTRACT FROM AUTHOR]

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup FN of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that FN has no minimal generating sets and present two particular infinite decreasing chains of generating sets of FN. [ABSTRACT FROM AUTHOR]

We obtain a criterion for the minimal logarithmic growth rate for an arbitrary set with a given set of operations defined on it, i.e., we describe all finite sets A with operations on them such that the growth rate differs by at most a constant from the logarithmic growth rate to base ∣A∣. [ABSTRACT FROM AUTHOR]

In recent years many papers have derived polyhedral and non-polyhedral convex envelopes for different classes of functions. Except for the univariate cases, all these classes of functions share the property that the generating set of their convex envelope is a subset of the border of the region over which the envelope is computed. In this paper we derive the convex envelope over a rectangular region for a class of functions which does not have this property, namely the class of bivariate cubic functions without univariate third-degree terms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the irreducible generating sets of orthogonal and symplectic rook monoids. We find that these irreducible generating sets are closely related to those of the unit groups of the monoids. We study and determine irreducible generating sets of order-preserving orthogonal rook monoids and order-preserving symplectic rook monoids. We then investigate nonirreducible generating sets of the singular parts of the orthogonal and symplectic rook monoids. [ABSTRACT FROM AUTHOR]

We introduce the notion of language uniform theory and study topological properties related to families of language uniform theory and their E-combinations. It is shown that the class of language uniform theories is broad enough. Sufficient conditions for the language similarity of language uniform theories are found. Properties of language domination and of infinite language domination are studied. A characterization for E-closure of a family of language uniform theories in terms of index sets is found. We consider the class of linearly ordered families of language uniform theories and apply that characterization for this special case. The properties of discrete and dense index sets are investigated: it is shown that a discrete index set produces a least generating set whereas a dense index set implies at least continuum many accumulation points and the closure without the least generating set. In particular, having a dense index set the theory of the E-combination does not have $e$-least models and it is not small. Using the dichotomy for discrete and dense index sets we solve the problem of the existence of least generating set with respect to $E$-combinations and characterize that existence in terms of orders. Values for $e$-spectra of families of language uniform theories are obtained. It is shown that any $e$-spectrum can be realized by $E$-combination of language uniform theories. Low estimations for $e$-spectra relative to cardinalities of language are found. It is shown that families of language uniform theories produce an arbitrary given Cantor-Bendixson rank and given degree with respect to this rank.

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories including the negation of finite character and the exchange property. It is shown that closure operators for E-combinations correspond to the closures with respect to the ultraproduct operator forming Hausdorff topological spaces. It is also shown that closure operators for disjoint $P$-combinations form topological $T_0$-spaces, which can be not Hausdorff. Thus topologies for $E$-combinations and $P$-combinations are rather different. We prove, for E-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set: it is shown that elements of the least generating set are isolated and dense in its $E$-closure. Related properties for P-combinations are considered: it is proved that again the existence of a minimal generating set of theories is equivalent to the existence of the least generating set but it is not equivalent to the isolation of elements in the generating set. It is shown that $P$-closures with the least generating sets are connected with families which are not $\omega$-reconstructible, as well as with families having finite $e$-spectra. Two questions on the least generating sets for E-combinations and P-combinations are formulated and partial answers are suggested.

In studying the structural properties of elementary theories, a relationship between theories with respect to a series of natural operators plays an important role. This relationship can be determined by placing models of given theories in various formula definable sets. Such sets include, for example, sets defined by unary predicates or equivalence relations. In this way, P-operators and E-operators arise, as well as their closures, and e-spectra, i.e. the numbers of new theories that may be generated by these operators. For E-operators, applicable to the families of theories of abelian groups, closures and generating sets, as well as their e-spectra are described. Szmielew invariants are used as a tool for the established characterization of a theory belonging to the E-closure of a family of theories of abelian groups. A series of families of theories corresponding to the sets of Szmielew invariants, properties of these families, and values of e-spectra are also described.

A subset X of a finite group G is called g-independent if there is no proper subset Y of X such that = (X, Φ(G)>. The group G has the embedding property if every g-independent subset of G can be embedded in a minimal generating set of G. If X is a set of prime power order elements, then we say that G has the pp-embedding property. In this note we classify all finite solvable groups with the pp-embedding property. Moreover we prove that this class is equal to the class of finite solvable groups with the embedding property. [ABSTRACT FROM AUTHOR]

Let Tn and Sn be the full transformation semigroup and the symmetric group on Xn = {1, ..., n}, respectively. For n, m ∈ Z+ with m ≤ n - 1 let T(n,m) = {α ∈ Tn : Xmα = Xm}. In this paper we research generating sets and the rank of T(n,m). In particular, we prove that ... for 1 ≤ m ≤ n - 1. [ABSTRACT FROM AUTHOR]

Let Tn and Sn be the full transformation semigroup and the symmetric group on Xn = {1, ..., n}, respectively. For n, m ∈ Z+ with m ≤ n - 1 let T(n,m) = {α ∈ Tn : Xmα = Xm}. In this paper we research generating sets and the rank of T(n,m). In particular, we prove that ... for 1 ≤ m ≤ n - 1. [ABSTRACT FROM AUTHOR]

We consider computational problems concerning algebras over finite fields. In particular, we propose an algorithm for finding a small generating set for the multiplicative group of F q [ x ] / F , where q = p n is a prime power and F ∈ F q [ x ] is a polynomial not necessarily irreducible. Based on this result, a new set of expander graphs can be explicitly constructed. In addition, we present algorithms for basis construction and decomposition of a given element with respect to the basis. [ABSTRACT FROM AUTHOR]

A finite lattice is interval dismantlable if it can be partitioned into an ideal and a filter, each of which can be partitioned into an ideal and a filter, etc., until you reach 1- element lattices. In this note, we find a quasi-equational basis for the pseudoquasivariety of interval dismantlable lattices, and show that there are infinitely many minimal interval non-dismantlable lattices. [ABSTRACT FROM AUTHOR]

We present two conjectures concerning the diameter of a direct power of a finite group. The first conjecture states that the diameter of Gn with respect to each generating set is at most n(|G|−rank(G)); and the second one states that there exists a generating set 𝒜, of minimum size, for Gn such that the diameter of Gn with respect to 𝒜 is at most n(|G|−rank(G)). We will establish evidence for each of the above mentioned conjectures. [ABSTRACT FROM AUTHOR]

In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes. [ABSTRACT FROM AUTHOR]