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Recently, Paiva et al. introduced the concept of quasi-overlap functions on bounded lattices and investigated some vital properties of them. In this paper, we continue consider this research topic and focus on the constructions of quasi-overlap functions along with their generalized forms on bounded partially ordered sets. To be specific, firstly, we generalize the truth values set of quasi-overlap functions from bounded lattices to bounded partially ordered sets and introduce the notions of 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P by considering the weaker boundary conditions than the quasi-overlap functions on P. Secondly, we give the constructions of quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P via the so-called Galois s-connections and 0,1-homomorphisms, 1-homomorphisms, 0-homomorphisms and ord-homomorphisms, respectively. In particular, we prove that those constructions contain the methods of extending the known quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions from any bounded partially ordered set P to any other bounded partially ordered sets. Finally, we show that those extensions maintain some basic properties of the known quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on P, such as, idempotent, Archimedean property and cancellation law.

The law of importation between fuzzy implications and conjunctions is an important property in both the theory and the application of fuzzy logic. Although many interesting results have been reported in the literature, the related open problem of finding all possible pairs of implications and conjunctions satisfying the law of importation has not been fully solved. As a continuation of two articles S. Massanet et al. (2018) [23] ; S. Massanet et al. (2018) [24] , this article aims to characterize fuzzy implication solutions with a continuous natural negation to the law of importation with respect to a fixed conjunctive uninorm having continuous underlying operators in several cases. Such solutions for the cases where the continuous underlying triangular norm and triangular conorm of a prefixed uninorm are idempotent or Archimedean are completely characterized, and the solutions for the cases where the continuous underlying operators are given by ordinal sums are characterized under some additional assumptions. Finally, some comments on relations to recent independent work W.-H. Li and F. Qin (2021) [25] are given. These two independent articles will go a further step towards answering the open problem regarding the law of importation.

As one class of binary continuous aggregation functions which play an important role in practical applications, overlap functions defined on the unit closed interval have been developed rapidly in the past decade. At the same time, Paiva et al. recently extended the concept of overlap functions on unit closed interval to the lattice-valued status and called them quasi-overlap functions on bounded lattices. In this paper, we mainly study the extension methods of quasi-overlap functions and their three generalized forms on bounded partially ordered sets. More concretely, first, we show some new extension methods of quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P by the so-called ⋄-operators and 0,1-homomorphisms, 1-⋄-operators and 1-homomorphisms, 0-⋄-operators and 0-homomorphisms, and 0,1-⋄-operators and ord-homomorphisms, respectively, which are different from the extension methods obtained by Qiao lately. And then, as an application of the new extension methods, some concrete quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on some certain bounded partially ordered set P are constructed. Finally, we prove that these extensions maintain idempotent and Archimedean property of the known quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P.

Recently, Zhang et al. studied the distributivity between uni-nullnorms and overlap (grouping) functions [68] . They obtained the sufficient and necessary conditions for the distributivity equations between them. However, the distributivity between null-uninorms and overlap (grouping) functions is missing. To fill this gap, in this paper, we explore some new results on the distributivity equations between overlap (grouping) functions and null-uninorms, which are the generalization of uninorms and nullnorms. Meanwhile, we give the full characterization of any idempotent null-uninorm.

Let $X$ be a finite connected poset, $F$ a field and $I(X,F)$ the incidence algebra of $X$ over $F$. We describe the bijective linear idempotent preservers $\varphi:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $\varphi$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $\varphi$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F=\mathbb{Z}_2$, then $\varphi$ is the composition of a bijective shift map and a Lie automorphism of $I(X,F)$. Under certain restrictions on the characteristic of $F$ we also obtain descriptions of the bijective linear maps which preserve tripotents and, more generally, $k$-potents of $I(X,F)$ for $k\ge 3$., Comment: Final version published in Linear Algebra and its Applications

The aim of this paper is to continue the work started in Pavlović (Filomat 30(14):3725–3731, 2016). We investigate further the properties of the local closure function and the spaces defined by it using common ideals, like ideals of finite sets, countable sets, closed and discrete sets, scattered sets and nowhere dense sets. Also, closure compatibility between the topology and the ideal, idempotency, and cases when the local closure of the whole space X is X or a proper subset of X, are closely investigated. In the case of closure compatibility and idempotency of the local closure function, the topology obtained by the local closure function is completely described. © 2021, Akadémiai Kiadó, Budapest, Hungary.

In this paper, construction methods of overlap and grouping functions on complete lattices via complete homomorphisms and complete 0 L , 1 L -endomorphisms are investigated. At first, we propose the notion of O-generator triple of overlap functions. Then, properties such as ( α , B , C ) -migrativity, ( B , C ) -homogeneity, idempotency and cancellation law for the overlap functions obtained by such generator triples are discussed. Finally, we give an analogous discussion on grouping functions on complete lattices.

The structure of idempotent n-uninorms is studied. We show that each idempotent 2-uninorm can be expressed as an ordinal sum of an idempotent uninorm (possibly also of a countable number of idempotent semigroups with operations min and max) and a 2-uninorm from Class 1 (possibly restricted to open or half-open unit square). Similar results are shown also for idempotent n-uninorms. Further, it is shown that idempotent n-uninorms are in one-to-one correspondence with special lower semi-lattices defined on the unit interval. The z-ordinal sum construction for partially ordered semigroups is also defined.

In 2018, Qiao and Hu [48] studied the α-migrativity of uninorms over overlap and grouping functions when the uninorm U belongs to one certain class (e.g., U min , U max , the family of idempotent uninorms, representable uninorms or uninorms continuous on ] 0 , 1 [ 2 ). In addition, they obtained some equivalent characterizations of them. This paper will continue to consider the characterizations of this kind of migrativity equations by means of the ordinal sum of overlap and grouping functions. We give the necessary and sufficient conditions for the solutions of the ( α , O ) -migrativity equation when the uninorm U becomes a t-norm or a conjunctive uninorm locally internal on the boundary and the ( α , G ) -migrativity equation when the uninorm U becomes a t-conorm or a disjunctive uninorm locally internal on the boundary, respectively.

In recent years, overlap functions, as a class of bivariate aggregation operators that are widely used in various application problems (see, e.g., in decision-making, image processing, classifications etc.), have been generalized to many forms. In particular, Paiva et al. (R. Paiva, E. Palmeira, R. Santiago, B. Bedregal, Lattice-valued overlap and quasi-overlap functions, Information Sciences 562 (2021) 180–199.) have generalized the overlap functions to the so-called quasi-overlap functions lately. In the meantime, considering aggregation operators on finite chains, especially the commonly bivariate aggregation operators (see, e.g., t-norms, t-conorms, uninorms, t-operators etc.) has become an important research topic in the fields of aggregation operators. In this paper, we take into account this research topic for quasi-overlap functions. First of all, we give the concept of quasi-overlap functions on a finite chain L with n + 2 elements and its arbitrary subchains together with three generalized forms of quasi-overlap functions on any subchain of L . And then, we show some examples of quasi-overlap functions on L along with some of its specific subchains and study the idempotent property, Archimedean property and cancellation law of quasi-overlap functions on L . Finally, we obtain two construction methods of quasi-overlap functions on L , one of them is the ordinal sum construction.

Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing only two distinct elements incomparable with a. Furthermore, some basic properties for the bounded lattice containing only two distinct elements incomparable with a are presented.

Let $$X_{n}=\{1, 2,\dots , n\}$$ where $$n\ge 2$$ . For $$a, b\in X_{n}$$ with $$a\ne b$$ , let $$a\atopwithdelims ()b$$ denote the idempotent transformation on $$X_{n}$$ which maps a to b and x to itself for $$x\ne a$$ . Let D be a simple digraph with vertex set $$X_{n}$$ and consider a set $$I(D) = \{ {a\atopwithdelims ()b} : \ D \ \text{ has } \text{ an } \text{ edge } \ b\rightarrow a \}.$$ We prove that the associated semigroup $$S(D)=\langle I(D)\rangle $$ intersects every $${{\mathcal {R}}}$$ -class of $$Sing_{n}$$ , the semigroup of singular transformations on $$X_{n}$$ , if and only if D is semicomplete. For any semicomplete digraph D, Green’s *-relations on the semigroup S(D) are described, and S(D) is shown to be abundant. We also find necessary and sufficient conditions for S(D) to be regular.

Let $$\mathbb {F}_q$$ be the finite field of order $$q=p^m$$ , where p is a prime, m is a positive integer, and $$\mathcal {R}=\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle $$ . Thus $$\mathcal {R}[x;\theta ,\delta _\theta ]$$ is a noncommutative ring, known as skew polynomial ring, where $$\theta $$ is an automorphism of $$\mathcal {R}$$ and $$\delta _\theta $$ is a $$\theta $$ -derivation of $$\mathcal {R}$$ . The main concern of this work is to characterize $$(\theta , \delta _\theta )$$ -cyclic codes over the ring $$\mathcal {R}$$ . Towards this, first we establish existence of the right division algorithm in $$\mathcal {R}[x;\theta ,\delta _\theta ]$$ . Then we find generating polynomials and idempotent generators for $$(\theta , \delta _\theta )$$ -cyclic codes over the ring $$\mathcal {R}$$ . Moreover, it is shown that $$(\theta , \delta _\theta )$$ -cyclic codes are principally generated. Finally, by using the decomposition method, we have provided several examples of $$(\theta , \delta _\theta )$$ -cyclic codes of different lengths over $$\mathcal {R}$$ out of them many are optimal as per the available database (Grassl, Code Tables: bounds on the parameters of various types of codes. http://www.codetables.de/ ).

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) The number of isomorphism classes of semisymmetric idempotent Latin squares of order $n$ equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order $n+1$, and (2) Suppose $A$ and $B$ are totally symmetric Latin squares of order $n\not\equiv0\bmod3$. If $A$ and $B$ are paratopic then $A$ and $B$ are isomorphic.

In this paper, we study ad-nilpotent elements of semiprime rings R with involution $$*$$ whose indices of ad-nilpotence differ on $${{\,\mathrm{Skew}\,}}(R,*)$$ and R. The existence of such an ad-nilpotent element a implies the existence of a GPI of R, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotients $$Q_m^s(R)$$ of R, a remains ad-nilpotent of the original indices in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ and $$Q_m^s(R)$$ . There exists an idempotent $$e\in Q_m^s(R)$$ that orthogonally decomposes $$a=ea+(1-e)a$$ and either ea and $$(1-e)a$$ are ad-nilpotent of the same index (in this case the index of ad-nilpotence of a in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ is congruent with 0 modulo 4), or ea and $$(1-e)a$$ have different indices of ad-nilpotence (in this case the index of ad-nilpotence of a in $${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$$ is congruent with 3 modulo 4). Furthermore, we show that $$Q_m^s(R)$$ has a finite $${\mathbb {Z}}$$ -grading induced by a $$*$$ -complete family of orthogonal idempotents and that $$eQ_m^s(R)e$$ , which contains ea, is isomorphic to a ring of matrices over its extended centroid. All this information is used to produce examples of these types of ad-nilpotent elements for any possible index of ad-nilpotence n.

We study those pre-Hilbert absolute-valued algebras satisfying the identity ( x 2 , y , x 2 ) = 0 . We prove that such an algebra A is finite-dimensional in each one of the following two cases: (1) A satisfies the additional identity ( x , x 2 , x ) = 0 , (2) A contains a weak left-unit. In the first case A is flexible and isomorphic to either R , C , C ⁎ , H , H ⁎ , O , O ⁎ or P . In the second one A has a left-unit and is isomorphic to either R , C , C ⁎ , H , H ⁎ , O , or O ⁎ . We also prove the existence of infinite-dimensional pre-Hilbert absolute-valued algebras satisfying ( x 2 , y , x 2 ) = 0 and containing only a non-zero idempotent.

In this article, we answer positively an open question posed in [G. Mayor, J. Suner, and J. Torrens, “Copula-like operations on finite settings,” IEEE Trans. Fuzzy Syst. , vol. 13, no. 4, pp. 468–477, Aug. 2005] concerning the extensions of discrete copulas to shuffles of Min. Moreover, we also use the extension of discrete copulas to sparse copulas, associated with discrete copulas, of idempotent copulas, in particular, the copula $M$ , in order to answer the open question in an alternative way.

In this paper, we investigate the full multidimensional moment problem for discrete measure using Vasilescu's idempotent approach based on Λ-multiplicative elements with respect to the associated square positive Riesz functional Λ. We give a sufficient condition for the existence of a discrete integral representation of the Riesz functional Λ, which turns out to be necessary in the bounded shift space case (in fact, it suffices to assume the density of polynomials in the corresponding L 2 -space). We pay special attention to Λ-multiplicative elements, providing several criteria guaranteeing that they are characteristic functions of single point sets. We also give an example showing that Λ-multiplicative elements may not be characteristic functions of single point sets.

There is a long history of studying Ramsey theory using the algebraic structure of the Stone–Cech compactification $$\beta S$$ of a discrete semigroup S. It has been shown that various Ramsey theoretic structures are contained in different algebraically large sets. In this article we deduce combinatorial characterizations of certain sets that are members of idempotent ultrafilters of closed subsemigroups of $$\beta S$$ , arising from certain Ramsey families. In the special case when $$S={\mathbb {N}}$$ , we deduce that sets which are members of all idempotent ultrafilters in these semigroups contain certain additive and multiplicative structures. We generalize this result for weak rings where we establish a non-commutative version of the additive and multiplicative structure.

Numerous variants have been proposed for sets of linguistic terms and the interval-valued hesitant fuzzy set (IVHFS). In particular, the interval-valued hesitant fuzzy linguistic set (IVHFLS) is more suitable for defining the hesitancy and inconsistency inherent in the human cognitive processes of decision making. A key aggregation operator is Heronian mean (HM), based on which the correlation among aggregated arguments can be captured. However, the existing HM operators partially overlook the correlation among more than two arguments and lack the properties of idempotency and reducibility. In this work, the limitations of HM operators are first analyzed. Then, two new HM variants are introduced: three-parameter weighted Heronian mean (TPWHM) and three-parameter weighted geometric Heronian mean (TPWGHM). Thus, the reducibility, idempotency, monotonicity, and boundedness properties are proven for the two computational procedures, and unique situations are mentioned. Furthermore, two more elaborate operators are also introduced which are called the interval-valued hesitant fuzzy linguistic TPWHM (IVHFLTPWHM) and the interval-valued hesitant fuzzy linguistic TPWGHM (IVHFLTPWGHM). The main properties, as well as unique situations of these two computational procedures, are discussed. Finally, the introduced methods are clarified by illustrative examples. In addition, the parameter effects on the decision-making outcomes are discussed and comparisons with other reference methods are made.

Hesitant fuzzy cognitive information provides an effective and convenient form for expressing human subjective cognition about the research object, i.e., hesitant fuzzy sets (HFSs). Studies on decision making with HFSs have become an important branch in decision theory, in which operational laws of hesitant fuzzy elements (HFEs) play a core role in the solution. However, current HFE computational laws have the disadvantages of subjectivity and dimensional problems. Therefore, how to define an objective HFE operational rule without dimensional problems is an open issue. This paper introduces an idempotent HFE computing rule to overcome current disadvantages. The weighted mean of HFEs under the developed computing rule is further discussed. In addition, a novel comparison law between HFEs is proposed. The property of idempotence is introduced to provide an intuitive integrated result of HFEs. To decrease the integrated HFEs dimensions, the sliding window model is utilized. Fundamental mathematical properties of the developed operations are discussed. Furthermore, the normal weighted means of HFEs are extended by using the developed idempotent computing rules. Finally, a novel comparison law for comparing HFEs is designed, which is further used to provide a multiattribute decision procedure. Additive idempotent is developed as a special and intuitive property for the HFE additive operation. Normal weighted means of HFEs, including arithmetic and geometric means, are correspondingly derived. Numerical examples have shown that the proposed HFE operational laws are valid, which can effectively decrease the dimensions of integrated results. The developed idempotent computing rules provide a novel HFE algebra structure, which includes the additive operation, multiplicative operation, scalar multiplication and power operation. By using the sliding window model, the developed idempotent computing rules can effectively reduce the integrated HFEs dimensions. The strength of the developed computational model is that integrating two identical pieces of cognitive information produces the same result. In addition, the modified HFE comparison law can overcome the drawback of current comparison laws, and a much more reasonable comparison result can be obtained.

Building on the previous extensive study of Yang, Gould and the present author, we provide a more precise insight into the group-theoretical ramifications of the word problem for free idempotent generated semigroups over finite biordered sets. We prove that such word problems are in fact equivalent to the problem of computing intersections of cosets of certain subgroups of direct products of maximal subgroups of the free idempotent generated semigroup in question, thus providing decidability of those word problems under group-theoretical assumptions related to the Howson property and the coset intersection property. We also provide a basic sketch of the global semigroup-theoretical structure of an arbitrary free idempotent generated semigroup, including the characterisation of Green's relations and the key parameters of non-regular $\mathscr{D}$-classes. In particular, we prove that all Sch\"utzenberger groups of $\mathsf{IG}(\mathcal{E})$ for a finite biordered set $\mathcal{E}$ must be among the divisors of the maximal subgroups of $\mathsf{IG}(\mathcal{E})$., Comment: Israel Journal of Mathematics (to appear). 28 pages; some minor text overlap with arXiv:1802.02420 and arXiv:1412.5167 in the preliminary section (because of the closely related topic)

In 2015, Mas et al. investigated the migrativity of uninorms and nullnorms over any fixed t-norm and t-conorm. First, they introduced the concept of migrative uninorm over any fixed t-norm. In addition, they showed equivalent characterizations of the migrativity equation when the uninorm belongs to one certain class (e.g., U min , U max , the family of idempotent uninorms, representable uninorms or uninorms continuous on ] 0 , 1 [ 2 ). And then, they gave the notion of migrative uninorms over any fixed t-conorm and proposed the migrativity equation using an analogous method. Finally, they discussed the migrativity of nullnorms over any fixed t-norm and t-conorm and obtained equivalent characterizations of the related migrativity equations, respectively. However, they did not discuss the migrativity property of a conjunctive (resp. disjunctive) uninorm non locally internal on the boundary over any fixed t-norm (resp. t-conorm) and the migrativity property of a conjunctive (resp. disjunctive) uninorm locally internal on the boundary over any continuous t-norm (resp. t-conorm). To fill the gap, in this paper, we explore some new results on the migrativity property of a conjunctive (resp. disjunctive) uninorm over any fixed t-norm (resp. t-conorm) and give some new characterizations of the migrativity property of nullnorms over any fixed t-norm and t-conorm, respectively.

We investigate situations in which compatible orders on rings with idempotent elements can be extended to more restrictive orders without assuming that the rings are commutative or that they contain multiplicative identities. We show that a ring with a certain kind of irreducible idempotent element is D∗ (i.e., every compatible partial order can be extended to a lattice order that makes the ring a d-ring) if and only if it is O∗ (i.e., every compatible partial order can be extended to a compatible total order); we characterize D∗-algebras over $\mathbb Q$ that have no identities and are O∗, and we find a class of D∗-algebras that contain subalgebras that are not D∗.

The Pythagorean fuzzy hypersoft set (PFHSS) is the most advanced extension of the intuitionistic fuzzy hypersoft set (IFHSS) and a suitable extension of the Pythagorean fuzzy soft set. In it, we discuss the parameterized family that contracts with the multi-subattributes of the parameters. The PFHSS is used to correctly assess insufficiencies, anxiety, and hesitancy in decision-making (DM). It is the most substantial notion for relating fuzzy data in the DM procedure, which can accommodate more uncertainty compared to available techniques considering membership and nonmembership values of each subattribute of given parameters. In this paper, we will present the operational laws for Pythagorean fuzzy hypersoft numbers (PFHSNs) and also some fundamental properties such as idempotency, boundedness, shift-invariance, and homogeneity for Pythagorean fuzzy hypersoft weighted average (PFHSWA) and Pythagorean fuzzy hypersoft weighted geometric (PFHSWG) operators. Furthermore, a novel multicriteria decision-making (MCDM) approach has been established utilizing presented aggregation operators (AOs) to resolve decision-making complications. To validate the useability and pragmatism of the settled technique, a brief comparative analysis has been conducted with some existing approaches.

Let $ p_1, p_2, p_3, q $ be distinct primes and $ m={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}$. In this paper, it is shown that the explicit expressions of primitive idempotents in the semi-simple ring $R_m = { F_q[x]}/{(x^m-1)}$ are the trace function of explicit expressions of primitive idempotents from $R_{p_i^{\alpha_i}}$. The minimal polynomials, generating polynomials and minimum distances of minimal cyclic codes of length $m$ over $F_q$ are also discussed. All the results obtained in \cite{ref[1]}, \cite{ref[4]}, \cite{ref[5]}, \cite{ref[6]}, \cite{ref[11]} and \cite{ref[14]} are simple corollaries to the results obtained in the paper.

We present a framework for constructing congruence closure modulo permutation equations, which extends the abstract congruence closure framework for handling permutation function symbols. Our framework also handles certain interpreted function symbols satisfying each of the following properties: idempotency (I), nilpotency (N), unit (U), I U U, or N U U. Moreover, it yields convergent rewrite systems corresponding to ground equations containing permutation function symbols. We show that congruence closure modulo a given finite set of permutation equations can be constructed in polynomial time using equational inference rules, allowing us to provide a polynomial time decision procedure for the word problem for a finite set of ground equations with a fixed set of permutation function symbols., Comment: In Proceedings SCSS 2021, arXiv:2109.02501

We give a description of the class of endomorphisms in the semiring UTMn(S) of upper triangular matrices over an additively idempotent semiring S. The endomorphisms α such that α(Eij), where Eij is...

In this paper we intend to describe generalized Lie-type derivations using, among other things, a generalization for alternative algebras of the result: “If $F:A\to A$ is a generalized Lie n-derivation associated with a Lie n-derivation D, then a linear map $H=F-D$ satisfies $H(p_n(x_1,x_2,\ldots ,x_n)) =p_n(H(x_1),x_2,\ldots ,x_n)$ for all $x_1,x_2,\ldots ,x_n\in A$ ”. Thus, if A is a unital alternative algebra with a nontrivial idempotent e1 satisfying certain conditions, then a generalized Lie-type derivation $F : A \rightarrow A$ is of the form $F(x) = \lambda x + \Xi(x)$ for all $x \in A$ , where $\lambda \in Z(A)$ and $\Xi : A \rightarrow A$ is a Lie-type derivation.

We consider a special class of idempotent of semisimple group algebras which we call essential. We give some criteria to decide when a primitive idempotent is essential; then we consider group algebras of cyclic group over finite fields, establish the number of essential idempotents in this case and find a relation among essential idempotents in different algebras. Finally we apply this ideas to coding theory and compute examples of codes with the best known weight.

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes