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In the first part of this paper we present a new family of finite bounded posets whose clones of monotone operations are not finitely generated. The proofs of these results are analogues of those in the famous paper of Tardos. Another interesting family of finite posets from the finite generability point of view is the family of locked crowns. To decide whether the clone of a locked crown where the crown is of at least six elements is finitely generated or not one needs to go beyond the scope of Tardos’s proof. Although our investigations are not conclusive in this direction, they led to the results in the second part of the paper. We call a monotone operation ascending if it is greater than or equal to some projection. We prove that the clones of bounded posets are generated by certain ascending idempotent monotone operations and the 0 and 1 constant operations. A consequence of this result is that if the clone of ascending idempotent operations of a finite bounded poset is finitely generated, then its clone is finitely generated as well. We provide an example of a half bounded finite poset whose clone of ascending idempotent operations is finitely generated but whose clone is not finitely generated. Another interesting consequence of our result is that if the clone of a finite bounded poset is finitely generated, then it has a three element generating set that consists of an ascending idempotent monotone operation and the 0 and 1 constant operations.

Let Ω be a symmetric generating set of a finite group Γ. Assume that (Γ,Ω)be such that Γ=〈Ω〉and Ω satisfies the two conditions C1: the identity element e∉Ω and C2: if a∈Ω, then a−1∈Ω. Given (Γ,Ω)satisfying C1and C2, define a Cayley graph G=Cay(Γ,Ω)with V(G)=Γ and E(G)={(x,y)a|x,y∈Γ,a∈Ωandy=xa}. When Γ=Zn=〈Ω〉, it is called as circulant graph and denoted by Cir(n,Ω). In this paper, we give a survey about the results on dominating sets in Cayley graphs and circulant graphs. Keywords: Cayley graph, Unitary Cayley graph, Dominating sets, Efficient dominating sets, E-Chain

Minimization of generating sets (sets of generators) indicated in the International Tables for Crystallography has been performed for tetragonal, trigonal, and hexagonal space groups. It is established that generating sets of some groups cannot be minimized to a set of minimally possible power (minimal set) for a given group, which suggests that the generating set used in the International Tables for Crystallography should be improved.

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 u...

In this paper, we study quasi-cyclic codes of index $1\frac {1}{2}$ and co-index 2m over $\mathbb {F}_{q}$ and their dual codes, where m is a positive integer, q is a power of an odd prime and $\gcd (m,q) = 1$. We characterize and determine the algebraic structure and the minimal generating set of quasi-cyclic codes of index $1\frac {1}{2}$ and co-index 2m over $\mathbb {F}_{q}$. We note that some optimal and good linear codes over $\mathbb {F}_{q}$ can be obtained from this class of codes. Furthermore, the algebraic structure of their dual codes is given.

A strongly generated gyrogroup is a gyrogroup that comes with a specific generating set invariant under its gyroautomorphisms, which may be viewed as a suitable generalization of a group. The achievement of constructing a word metric on a strongly generated gyrogroup, together with the recent notion of a gyrogroup action, prompts us to extend the Svarc–Milnor lemma (also known as the fundamental lemma of geometric group theory) to the case of gyrogroups.

A lattice is $(1+1+2)$-generated if it has a four-element generating set such that exactly two of the four generators are comparable. We prove that the lattice Quo$(n)$ of all quasiorders (also known as preorders) of an $n$-element set is $(1+1+2)$-generated for $n=3$ (trivially), $n=6$ (when Quo(6) consists of $209\,527$ elements), n=11, and for every natural number $n\geq 13$. In 2017, the second author and J. Kulin proved that Quo$(n)$ is $(1+1+2)$-generated if either $n$ is odd and at least $13$ or $n$ is even and at least $56$. Compared to the 2017 result, this paper presents twenty-four new numbers $n$ such that Quo$(n)$ is $(1+1+2)$-generated. Except for Quo(6), an extension of Z\'adori's method is used., Comment: 14 pages, 4 figures

Let G be a Garside group endowed with the generating set $${\cal S}$$ of non-trivial simple elements, and let H be a parabolic subgroup of G. We determine a transversal T of H in G such that each θ ∈ T is of minimal length in its right-coset, Hθ, for the word length with respect to $${\cal S}$$ . We show that there exists a regular language L on $${\cal S} \cup {{\cal S}^{- 1}}$$ and a bijection ev: L → T satisfying $$\lg \left(U \right) = {\lg _{\cal S}}\left({ev\left(U \right)} \right)$$ for all U ∈ L. From this we deduce that the coset growth series of H in G is rational. Finally, we show that G has fellow projections on H but does not have bounded projections on H.

Let [Formula: see text] be the monoid of all orientation-preserving and extensive full transformations on [Formula: see text] ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of [Formula: see text], and so the rank and the idempotent rank of [Formula: see text] are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of [Formula: see text] and completely obtain their classifications.

For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $\Phi(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $\Phi(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $\Phi(G;C)$ contains the Frattini subgroup $\Phi(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $Aut(G;C)$ acts transitively on the pairs $\{c,c^{-1}\}$ from $C$. We show that, for a normal edge-transitive Cayley graph $Cay(G,C)$, its quotient modulo $\Phi(G;C)$ is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever the subgroup $\Phi(G;C)$ is trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all $4$-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi., Comment: 16 pages

Assume G is a finite group, such that |G| = 6pq or 7pq, where p and q are distinct prime numbers, and let S be a generating set of G. We prove there is a Hamiltonian cycle in the corresponding Cayley graph Cay(G;S).

Let $A\neq A_1, A_2, I_{2m}$ be an irreducible Artin--Tits group of spherical type. We show that periodic elements of $A$ and the elements preserving some parabolic subgroup of $A$ act elliptically on the additional length graph $\mathcal{C}_{AL}(A)$, an hyperbolic, infinite diameter graph associated to $A$ constructed by Calvez and Wiest to show that $A/Z(A)$ is acylindrically hyperbolic. We use these results to find an element $g\in A$ such that $\langle P,g \rangle\cong P* \langle g \rangle$ for every proper standard parabolic subgroup $P$ of $A$. The length of $g$ is uniformly bounded with respect to the Garside generators, independently of $A$. This allows us to show that, in contrast with the Artin generators case, the sequence $\{\omega(A_n,\mathcal{S})\}_{n\in \mathbb{N}}$ of exponential growth rates of braid groups with respect to the Garside generating set, goes to infinity., Comment: 19 pages, 2 figures

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.

We consider the graph $\Gamma _{\text {virt}}(G)$ whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

The 15-Puzzle is a well studied permutation puzzle. This paper explores the group structure of a three-dimensional variant of the 15-Puzzle known as the Varikon Box, with the goal of providing a heuristic that would help a human solve it while minimizing the number of moves. First, we show by a parity argument which configurations of the puzzle are reachable. We define a generating set based on the three dimensions of movement, which generates a group that acts on the puzzle configurations, and we explore the structure of this group. Finally, we show a heuristic for solving the puzzle by writing an element of the symmetry group as a word in terms of a generating set, and we compute the shortest possible word for each puzzle configuration.

We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $\lambda$ (a partition of $n$) is the number of multiset tableaux of shape $\lambda$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.

Let $N_g$ be the non-orientable surface of genus $g$, $\mathrm{MCG}(N_g)$ the mapping class group of $N_g$, $\mathscr{T} (N_g)$ the index $2$ subgroup generated by all Dehn twists of $\mathrm{MCG}(N_g)$. We prove that for odd genus, $(1)$ if $g = 4k + 3 (k\ge1), \mathrm{MCG}(N_g)$ can be generated by three elements of finite order; $(2)$ if $g = 4k + 1 (k\ge2)$, $\mathscr{T} (N_g)$ can be generated by three elements of finite order.

Let G be the classical group, and let Hom ( Z m , G ) denote the space of commuting m-tuples in G. First, we refine the formula for the Poincare series of Hom ( Z m , G ) due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincare series, and apply it to prove the dependence of the topology of Hom ( Z m , G ) on the parity of m and the rational hyperbolicity of Hom ( Z m , G ) for m ≥ 2 . Next, we give a minimal generating set of the cohomology of Hom ( Z m , G ) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom ( Z m , G ) with the best possible stable range. Baird proved that the cohomology of Hom ( Z m , G ) is identified with a certain ring of invariants of the Weyl group of G, and our approach is a direct calculation of this ring of invariants.

In this note we present explicit canonical forms for all the elements in the two-qubit CNOT-Dihedral group, with minimal numbers of controlled-S (CS) and controlled-X (CX) gates, using the generating set of quantum gates [X, T, CX, CS]. We provide an algorithm to successively construct the n-qubit CNOT-Dihedral group, asserting an optimal number of controlled-X (CX) gates. These results are needed to estimate gate errors via non-Clifford randomized benchmarking and may have further applications to circuit optimization over fault-tolerant gate sets., 9 pages. Accepted to Quantum on 30 Nov 2020

We derive a minimal generating set of planar moves for diagrams of surfaces embedded in the four-space. These diagrams appear as the bonded classical unlink diagrams., 13 pages, final vesion. arXiv admin note: text overlap with arXiv:1808.07652

A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. A system of elements of a free algebra is primitive if there is a complement of this system with respect to a free generating set of the free algebra. An element of a free algebra of a Schreier variety is said to be almost primitive if it is not primitive in the free algebra, but it is a primitive element of any subalgebra that contains it. This survey article is devoted to the study of primitive and almost primitive elements of Schreier varieties.

This paper is about two impartial combinatorial games, namely achievement and avoidance games. These games can be played on groups by two persons. If G is a finite group, then both players chose elements of G alternately. The player who builds a generating set for G is called the loser in avoidance game whereas the builder of generating set is called the winner in achievement game. Dana and Sieben have been provided concept of nim-numbers by plotting their structure diagrams for both games. Since, the nim-numbers provide a more easier analysis of impartial combinatorial games and provide final outcome of these impartial games. But it is hard to calculate the nim-numbers of an impartial game. In this paper, the nim-numbers for achievement and avoidance games on some non-abelian groups have been calculated. In particular, the simplified structure diagrams and nim-values of these two games for finite non-abelian p-groups, dicyclic type groups, and groups of the form ℂ3 × ℂp has been determined.

For a monoid $M$ and a subsemigroup $S$ of the full transformation semigroup $T_n$, the wreath product $M\wr S$ is defined to be the semidirect product $M^n\rtimes S$, with the coordinatewise action of $S$ on $M^n$. The full wreath product $M\wr T_n$ is isomorphic to the endomorphism monoid of the free $M$-act on $n$ generators. Here, we are particularly interested in the case that $S=Sing_n$ is the singular part of $T_n$, consisting of all non-invertible transformations. Our main results are presentations for $M\wr Sing_n$ in terms of certain natural generating sets, and we prove these via general results on semidirect products and wreath products. We re-prove a classical result of Bulman-Fleming that $M\wr Sing_n$ is idempotent generated if and only if the set $M/L$ of $L$-classes of $M$ forms a chain under the usual ordering of $L$-classes, and we give a presentation for $M\wr Sing_n$ in terms of idempotent generators for such a monoid $M$. Among other results, we also give estimates for the minimal size of a generating set for $M\wr Sing_n$, as well as exact values in some cases (including the case that $M$ is finite and $M/L$ is a chain, in which case we also calculate the minimal size of an idempotent generating set). As an application of our results, we obtain a presentation (with idempotent generators) for the idempotent generated subsemigroup of the endomorphism monoid of a uniform partition of a finite set., Comment: V2: 34 pages, 5 figures - reformatted and footnotes added concerning semigroups with a right identity. V1: 35 pages, 5 figures

Experience gained from studies of the transient processes that accompany torsional vibrations, including an analytic solution of the problem along with results of calculations of transient processes, is generalized. Features in the relationship between the driving torque of the turbine and the elastic moment on the shaft of the generating unit as a function of the point of application of jump in the load on the set are presented.

For an integer $n\geq 5$, H. Strietz (1975) and L. Z\'adori (1986) proved that the lattice Part$(n)$ of all partitions of $\{1,2,\dots,n\}$ is four-generated. Developing L. Z\'adori's particularly elegant construction further, we prove that even the $k$-th direct power Part$(n)^k$ of Part$(n)$ is four-generated for many but only finitely many exponents $k$. E.g., Part$(n)^k$ is four-generated for every $k\leq 3\cdot 10^{89}$, and it has a four element generating set that is not an antichain for every $k\leq 1.4\cdot 10^{34}$. In connection with these results, we outline a protocol how to use these lattices in authentication and secret key cryptography., Comment: 20 pages, 5 figures

The paper presents a method for measuring AC voltage and a simple sensor for the control system of an autonomous generating set. The sensor speed is a half the period of the generated voltage, the output signal does not depend on the frequency and interference.

The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese 3-fold projections. More precisely, for any integer d ≥ 4 and any d -th root e of 1 we denote by X d the toric variety defined as the image of the morphism φ T d : P 3 ⟶ P μ ( T d ) − 1 where T d are all monomials of degree d in k [ x , y , z , t ] invariant under the action of the diagonal matrix M ( 1 , e , e 2 , e 3 ) . In this work, we describe a Z -basis of the lattice L η associated to I ( X d ) as well as a minimal binomial set of generators of the lattice ideal I ( X d ) = I + ( η ) .

Minimal generating sets for 17 2D-space (plane) groups have been derived. The concept of fuzzy generating set of a group is introduced. The considered groups are compared with respect to the degree of fuzziness of their generating sets.

All possible minimal sets of generating elements of crystallographic point groups have been derived. The variety of generating sets for each abstractly distinguishable group is presented in the form of a fuzzy generating set. The degree of fuzziness of generating set is calculated, and a cluster analysis of crystal classes with respect to the degree of fuzziness is performed.

In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the finite chains. We found uniquely irreducible generating set for the given semigroups. Keywords: Semigroup, Semilattice, Binary relation

The neural ideal of a binary code C ⊆ F 2 n is an ideal in F 2 [ x 1 , … , x n ] closely related to the vanishing ideal of C . The neural ideal, first introduced by Curto et al., provides an algebraic way to extract geometric properties of realizations of binary codes. In this paper we investigate homomorphisms between polynomial rings F 2 [ x 1 , … , x n ] which preserve all neural ideals. We show that all such homomorphisms can be decomposed into a composition of three basic types. Using this decomposition, we can interpret how these homomorphisms act on the underlying binary codes. We can also determine their effect on geometric realizations of these codes using sets in R d . We also describe how these homomorphisms affect a canonical generating set for neural ideals, yielding an efficient method for computing these generators in some cases.

The main objective of this paper is to present an improved algorithm to compute H-bases by relying on techniques from linear algebra. An H-basis is a specific generating set for a polynomial ideal ...

To any toric ideal I A , encoded by an integer matrix A, we associate a matroid structure called the bouquet graph of A and introduce another toric ideal called the bouquet ideal of A. We show how these objects capture the essential combinatorial and algebraic information about I A . Passing from the toric ideal to its bouquet ideal via the graph theoretic properties of the bouquet graph allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call stable, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie toric ideals whose various bases (e.g., minimal generating sets, Grobner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Grobner basis, any reduced Grobner basis and any minimal generating set coincide.

In a recent paper Botur et al. (2018) we have presented a generating set of the clone of idempotent aggregation functions on bounded lattices . As the main result we have shown that this clone is generated by certain ternary idempotent functions from which all idempotent aggregation functions of L can be obtained by usual term composition. The aim of this paper is to present an essential improvement of the result above by presenting a new generating set of this clone. A bit artificial ternary functions are substituted here by natural (binary) lattice a-medians and certain binary characteristic functions. Consequently, the clone is generated by its binary part and the result strengthens the essential role of medians within all idempotent aggregation functions. Moreover, we will show that for an n-element lattice L, the upper bound of binary generators is 2 n − 1 .

This paper considers the degree-diameter problem for undirected circulant graphs. For degrees 10 and 11, newly discovered families of circulant graphs of arbitrary diameter are presented which are largest known and are conjectured to be extremal. They are also the largest-known Abelian Cayley graphs of these degrees. For each such family, the order of every graph in the family is defined by a quintic polynomial function of the diameter which is specific to the family. The elements of the generating set for each graph are similarly defined by a set of polynomials in the diameter. The existence of the graphs in the degree 10 families has been proved for all diameters. These graphs are consistent with a conjecture on the order of extremal Abelian Cayley and circulant graphs of any degree and diameter.

By using quantum vertex operators we study the invariance of the rank n free-fermion vertex algebra under the action of the group ℤ∕2ℤ and obtain its minimal generating set. When n = 1, it is well ...

In this work we determine the so-called minimal generating set of the Weierstrass semigroup of certain m points on curves X with plane model of the form f ( y ) = g ( x ) over F q , where f ( T ) , g ( T ) ∈ F q [ T ] . Our results were obtained using the concept of discrepancy, for given points P and Q on X . This concept was introduced by Duursma and Park, in [5] , and allows us to make a different and more general approach than that used to certain specific curves studied earlier.

Let Γ n be the symmetric group on { 1 , 2 , … , n } and S be the generating set of Γ n . The corresponding Cayley graph is denoted by Γ n ( S ) . If all elements of S are transpositions, a simple way to depict S is via a graph, called the transposition generating graph of S , denoted by A ( S ) (or say simply A ), where the vertex set of A is { 1 , 2 , … , n } , there is an edge in A between i and j if and only if the transposition ( i j ) ∈ S , and Γ n ( S ) is called a Cayley graph obtained from a transposition generating graph A . In this paper, by exploring and utilizing the structural properties of these Cayley graphs, we obtain that the pessimistic diagnosability of Γ n ( S ) is equal to 2 | E ( A ) | − 2 if A has no triangles or 2 | E ( A ) | − 3 if A has a triangle. As corollaries, the pessimistic diagnosability of many kinds of graphs such as Cayley graphs generated by unicyclic graphs, wheel graphs, complete graphs, and tree graphs is obtained.

We study a q-player variation of the impartial avoidance game introduced by Anderson and Harary, where q is a prime. The game is played by the q players taking turns selecting previously-unselected elements of a finite group. The losing player is the one who selects an element that causes the set of jointly-selected elements to be a generating set for the group, with the previous player winning. We introduce a ranking system for the other players to prevent coalitions. We describe the winning strategy for these games on cyclic, nilpotent, dihedral, and dicyclic groups.

We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer-Vietoris spectral sequences for configuration spaces., Comment: 21 pages, 7 figures. Revised Proposition 2.3, other minor changes

Let M n ( F ) be the algebra of n × n matrices over the field F and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M n ( F ) cannot exceed 2 n − 2 . We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2 n − 2 is achieved.

In this paper we examine the descriptive potential of a combinatorial data structure known as "Generating Set" in constructing the boundary maps of a simplicial complex. By refining the approach of \cite{Dumas} in generating these maps, we provide algorithms that allow for relations among simplices to be easily accounted for. In this way we explicitly generate faces of a complex only once, even if a face is shared among multiple simplices. The result is a useful interface for constructing complexes with many relations and for extending our algorithms to $\Delta$-complexes. Once we efficiently retrieve the representatives of "living" simplices i.e., of those that have not been related away, the construction of the boundary maps scales well with the number of relations and provides a simpler alternative to JPlex. We finish by noting that the generating data of a complex is equivalent in information to its incidence matrix and we provide efficient algorithms for converting from an incidence matrix to a Generating Set.

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.

In a recent paper we proposed the study of aggregation functions on lattices via clone theory approach. Observing that aggregation functions on lattices just correspond to $0,1$-monotone clones, we have shown that all aggregation functions on a finite lattice $L$ can be obtained as usual composition of lattice operations $\wedge,\vee$, and certain unary and binary aggregation functions. The aim of this paper is to present a generating set for the class of intermediate (or, equivalently, idempotent) aggregation functions. This set consists of lattice operations and certain ternary idempotent aggregation functions., Comment: 18 pages

In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the class Σ8 (X, n + k +1) , and find uniquely irreducible generating set for the given semigroups.

Let $G =\langle S\rangle$ be a solvable permutation group given as input by the generating set $S$, that is, $G$ is a solvable subgroup of the symmetric group $S_n$. We give a deterministic polynom...

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.