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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.

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

In 2009, Ted and Paul Hurley proposed a code construction method using group rings. These codes with single generator are termed group ring codes and in particular zero-divisor codes when using zero-divisors as generators. In this paper, we mainly study the equivalency of zero-divisor codes in $$F_2G$$ having generator from I(G), the set of all idempotents in $$F_2G$$ . For abelian G, our previous notion of generated idempotents completely classified I(G) by serving as its basis. Here, we first extend the notion of generated idempotents to study and classify some elements in I(G) for non-abelian G. Later, the study is generally done on equivalency of zero-divisor codes in $$F_2G$$ , then concentrating on those with idempotent generator. In particular, we affirm the conjecture “Every group ring code in $$F_2D_{2n}$$ is equivalent to some in $$F_2C_{2n}$$ ” in the cases where the generators are our classified idempotents. We also show that the equivalency of zero-divisor codes in $$F_2C_n$$ with generated idempotent as generators can be established sufficiently on the generator property.

The RSA cryptosystem [14] requires two primes (p, q). Its security is based on the time required to factor n = pq without knowing p or q, since no polynomial-time factoring algorithms are known. Cu...

We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this formula to obtain the stabilization algebras $\dot{\mathbf K}^{\mathfrak{c}}_n$, $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$, $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \imath}_{\eta}$, which are idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ with compatible monomial, standard and canonical bases., Comment: 39 pages, fixed some typos, to appear in Journal of Pure and Applied Algebra

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized ( max ⁡ , + ) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.

Costas Latin squares are important combinatorial structures in combinatorial design theory. Some Costas Latin squares are found in recent years, but there are still some open problems about the existence of Costas Latin squares with specified properties including idempotency, orthogonality, and certain quasigroup properties. In this paper, we describe an efficient method for solving these problems using state-of-the-art SAT solvers. We present new results of Costas Latin squares with specified properties of even order \(n \le 10\). It is found that within this order range, most Costas Latin squares with such properties don’t exist except for a few cases. The non-existence can be certified since SAT solvers can produce a formal proof. Experimental results demonstrate the effectiveness of our method.

This paper investigates the computational complexity of deciding if a given finite idempotent algebra has a ternary term operation $m$ that satisfies the minority equations $m(y,x,x)\approx m(x,y,x)\approx m(x,x,y)\approx y$. We show that a common polynomial-time approach to testing for this type of condition will not work in this case and that this decision problem lies in the class NP.

The concept of quantization dimensions (upper and lower) of idempotent probability measures on a metric compactum is introduced. These dimensions, which are defined using a general functorial scheme, are analogous to the quantization dimensions of classical probability distributions. It is proved that the quantization dimensions of an idempotent measure do not exceed the corresponding box dimensions of its support. At the same time, for any non-negative number b not exceeding the upper box dimension of a metric compactum X, there exists an idempotent probability measure μ b with support equal to X and the upper quantization dimension equal to b. For the lower quantization dimension, a similar intermediate value theorem is valid under some additional restrictions on the compactum X.

We say that a category $\mathscr{D}$ is dimension zero over a field $F$ provided that every finitely generated representation of $\mathscr{D}$ over $F$ is finite length. We show that $\textrm{Rel}(R)$, a category that arises naturally from a finite idempotent semiring $R$, is dimension zero over any infinite field. One special case of this result is that $\textrm{Rel}$, the category of finite sets with relations, is dimension zero over any infinite field., Comment: updates for JOA submission

Let R be the Galois ring G R ( p k , s ) of characteristic p k and cardinality p s k . Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length n over R , and a p -adic integer ring with gcd ( p , n ) = 1 . Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length r l m over R , where r , l , and t are three primes with 2 ∤ l , r | ( q t − 1 ) , l v ∥ ( q t − 1 ) and gcd ( r l , q ( q − 1 ) ) = 1 . Finally, as applications, weight distributions of all irreducible cyclic codes for t = 2 and generator polynomials of self-dual cyclic codes of length l m and r l m over R are given.

We extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model. The semantics are based on fixing a subclass of the sets in a structure for the basic language that corresponds to the intended ultrafilter. In this language we state axioms that express the notion that the subclass is an ultrafilter and additional ones that say it is idempotent or Ramsey. The axioms for idempotent ultrafilters prove, for example, Hindman’s theorem and its generalizations such as the Galvin--Glazer theorem and iterated versions of these theorems (IHT and IGG). We prove that adding these axioms to IHT produce conservative extensions of ACA0+IHT,${\rm{ACA}}_{\rm{0}}^ +$, ATR0,${\rm{\Pi }}_2^1$-CA0, and${\rm{\Pi }}_2^1$-CA0for all sentences of second order arithmetic and for full Z2for the class of${\rm{\Pi }}_4^1$sentences. We also generalize and strengthen a metamathematical result of Wang (1984) to show, for example, that any${\rm{\Pi }}_2^1$theorem ∀X∃YΘ(X,Y) provable in ACA0or${\rm{ACA}}_{\rm{0}}^ +$there aree,k∈ ℕ such that ACA0or${\rm{ACA}}_{\rm{0}}^ +$proves that ∀X(Θ(X, Φe(J(k)(X))) where Φeis theeth Turing reduction andJ(k)is thekth iterate of the Turing or Arithmetic jump, respectively. (A similar result is derived for${\rm{\Pi }}_3^1$theorems of${\rm{\Pi }}_1^1$-CA0and the hyperjump.)

Simple cycles, also known as self-avoiding polygons, are cycles on graphs which are not allowed to visit any vertex more than once. We present an exact formula for enumerating the simple cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from an Hopf algebra, which we construct explicitly, and which provides further means of counting simple cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate simple cycles.

The uninorms with continuous underlying functions were characterized by their set of discontinuity points in the previous work of the author, using the characterizing set-valued function. In this paper properties of this characterizing set-valued function are studied. It is shown that the type of the monotonicity of such a set-valued function is always changed in idempotent points of the corresponding uninorm. Several additional properties of the characterizing set-valued function of a uninorm with continuous underlying functions are shown and an example is included. Results shown in this paper are used for a complete characterization of uninorms with continuous underlying functions via the ordinal sum construction, which is shown in the consecutive paper.

Although it is well known that Fredholmness of the linear combination α P + β Q , α , β ∈ C ∖ { 0 } , α + β ≠ 0 , does not depend on the choice of scalars if P , Q ∈ B ( H ) are idempotents, no necessary and sufficient conditions for Fredholmness of this linear combination are yet known, except for the special case when P and Q are orthogonal projectors. In this paper, using a completely different approach and some results on completion problems of operator matrices, we give necessary and sufficient conditions for Fredholmness of a sum of two idempotents. Also, we will discuss the more general question when the sum of two operators is a Fredholm operator and consider some special cases when Fredholmness of a linear combination of two operators is independent of the choice of the scalars. On the other hand some special classes of operators for which a linear combination of two operators depends of the choice of the scalars are listed. A new proof of a well-known result is given.

It is known that bisymmetry generalizes the simultaneous commutativity and associativity in the framework of the unit interval. In this work, we will completely characterize two classes of bisymmetric aggregation operators: one with a neutral element and the other with the vertical and horizontal sections of the idempotent elements being smooth on a finite chain, but not necessarily smooth and commutative. Thus, the previous results, based on the smoothness that is known as a very restrictive condition, are improved. For example, there is only one smooth Archimedean t-norm on a finite chain. In this paper, the discrete bisymmetric aggregation operators are explored without the limit of the smoothness. As a by-product, it is deduced that for smooth aggregation operators on a finite chain, the bisymmetry is equivalent to the commutativity and associativity, which improves the conclusion obtained by Mas et al. that associativity and bisymmetry are equivalent for commutative smooth aggregation operators on a finite chain.

Overlap and grouping functions have been proposed by Bustince et al.. In this paper, we continue investigating the two functions by their multiplicative generator pairs. At first, we introduce the concept of multiplicative generator pair for overlap functions. And then, we investigate the migrativity, homogeneity, idempotency, Archimedean and cancellation properties for the overlap functions obtained by such multiplicative generator pairs. Finally, we give the definition of multiplicative generator pair for grouping functions and show some related results by the duality of overlap and grouping functions.

Binary operations have many applications in fuzzy set theory. One of them is uninorms. In this paper, we study some properties of idempotent uninorms on bounded lattices. It is shown that idempotent uninorms on an arbitrary bounded lattice need not always be internal (with the extended definition of the term ”internal”). We give a sufficient condition for the bounded lattice to any idempotent uninorm be internal. In addition that it is given many properties of idempotent uninorms in this case.

A quasigroup ( Q , ⋅ ) of order v satisfies Stein's third law if ( y ⋅ x ) ⋅ ( x ⋅ y ) = x holds for all x, y ∈ Q . Let the quasigroup contain n idempotent elements. We construct such quasigroups with ( v , n ) ∈ { ( 20 , 0 ) , ( 24 , 0 ) , ( 28 , 0 ) , ( 36 , 0 ) } , thus completing the existence spectrum of quasigroups satisfying Stein's third law with no idempotents. We also construct previously unknown quasigroups with ( v , n ) ∈ { ( 17 , 11 ) , ( 21 , 3 ) , ( 21 , 7 ) , ( 24 , 4 ) , ( 25 , 7 ) , ( 25 , 19 ) } and provide an enumeration for all v ≤ 9 .

Let K be a field of characteristic zero, $${\mathcal {A}}$$ a K-algebra and $$\delta $$ a K-derivation of $${\mathcal {A}}$$ or K- $${\mathcal {E}}$$ -derivation of $${\mathcal {A}}$$ (i.e., $$\delta =\mathrm{Id}_{\mathcal {A}}-\phi $$ for some K-algebra endomorphism $$\phi $$ of $${\mathcal {A}}$$ ). Motivated by the Idempotent conjecture proposed in Zhao (Commun Contemp Math, https://doi.org/10.1142/S0219199717500560 , arXiv:1701.05992 ), we first show that for every idempotent e lying in both the kernel $${\mathcal {A}}^\delta $$ and the image $$\mathrm{Im}\,\delta {:}{=}\delta ({\mathcal {A}})$$ of $$\delta $$ , the principal ideal $$(e)\subseteq \mathrm{Im}\,\delta $$ if $$\delta $$ is a locally finite K-derivation or a locally nilpotent K- $${\mathcal {E}}$$ -derivation of $${\mathcal {A}}$$ ; and $$e{\mathcal {A}}, {\mathcal {A}}e \subseteq \mathrm{Im}\,\delta $$ if $$\delta $$ is a locally finite K- $${\mathcal {E}}$$ -derivation of $${\mathcal {A}}$$ . Consequently, the Idempotent conjecture holds for all locally finite K-derivations and all locally nilpotent K- $${\mathcal {E}}$$ -derivations of $${\mathcal {A}}$$ . We then show that (if and) only if $$\delta $$ is surjective, which generalizes the same result due to Nouaze and Gabriel (J Algebra 6(1):77–99, 1967) and Wright (Illinois J Math 25(3):423–440, 1981) for locally nilpotent K-derivations of commutative K-algebras to locally finite K-derivations and K- $${\mathcal {E}}$$ -derivations $$\delta $$ of all K-algebras $${\mathcal {A}}$$ .

Let S be a subsemigroup of a semigroup T and let IG(E) and IG(F) be the free idempotent generated semigroups over the biordered sets of idempotents of E of S and F of T, respectively. We examine th...

Cyclic structures on convolutional codes are modeled using an Ore extension A [ z ; σ ] of a finite semisimple algebra A over a finite field F . In this context, the separability of the ring extension F [ z ] ⊂ A [ z ; σ ] implies that every ideal code is a split ideal code. We characterize this separability by means of σ being a separable automorphism of the F -algebra A. We design an algorithm that decides if such a given automorphism σ is separable. In addition, it also computes a separability element of F [ z ] ⊂ A [ z ; σ ] , which is important because it can be used to find an idempotent generator of each ideal code with sentence-ambient A [ z ; σ ] .

We propose a new type of weighted averaging functions given implicitly as a solution to an algebraic equation. The main ingredient is incorporation of weights performed by using a function different to the product. Strict triangular norms are one example of such functions. We show that the weights should be normalized in this case differently to the usual sum to one condition. We establish the conditions under which the implicit averages are well defined, are idempotent, monotone and bounded by the minimum and maximum of their arguments. We present many examples of implicit averages, some of which recover the existing families of functions while others define new families. In particular we provide new generalizations of power means, density based averages and weighted OWA functions.

In this paper, we show that for a positive operator A on a Hilbert $$C^*$$ -module $$ \mathscr {E} $$ , the range $$ \mathscr {R}(A) $$ of A is closed if and only if $$ \mathscr {R}(A^\alpha ) $$ is closed for all $$\alpha \in (0,1)\cup (1,+\,\infty )$$ , and this occurs if and only if $$ \mathscr {R}(A)=\mathscr {R}(A^\alpha ) $$ for all $$\alpha \in (0,1)\cup (1,+\,\infty )$$ . As an application, we prove that for an adjontable operator A if $$\mathscr {R}(A)$$ is nonclosed, then $$\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty $$ . Finally, we show that for an adjointable operator A if $$ \overline{\mathscr {R}(A^*) } $$ is orthogonally complemented in $$ \mathscr {E} $$ , then under certain coditions there exists an idempotent C and a unique operator X such that $$ XAX=X, AXA=CA, AX=C $$ and $$ XA=P_{A^*} $$ , where $$ P_{A^*} $$ is the orthogonal projection of $$ \mathscr {E} $$ onto $$ \overline{\mathscr {R}(A^*)}$$ .

In this paper, π-regular (resp. strongly π-regular) and idempotent elements in a ring is used to define π-clean (resp. strongly π-clean) rings. A ring is called π-clean (resp. strongly π-clean) if each of its elements is the sum of a π-regular (resp. strongly π-regular) and an idempotent element. The purpose of this research is to give some results on π-clean and strongly π-clean rings and study π-clean and strongly π-clean group rings. Also, this paper is involved with investigating F 2 C 2 as a social group and measuring influence a member of it’s rather than others.

In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

For $$n\in \mathbb {N}$$ , let $$O_{n}$$ be the semigroup of all order-preserving transformations on the finite chain $$X_{n}=\{1,\ldots ,n\}$$ , under its natural order. For any non-empty subset A of $$X_{n}$$ , let $$O_{n}(A)$$ and $$O_{n}^+(A)$$ be the subsemigroups of all order-preserving and A-decreasing, and of all order-preserving and A-increasing transformations on $$X_{n}$$ , respectively. In this paper we obtain formulae for the number of elements and for the number of idempotents in $$O_{n}(A)$$ . Moreover, we show that $$O_{n}(A)$$ contains a zero element if and only if $$1\in A$$ , and then we obtain the number of nilpotents in $$O_{n}(A)$$ when $$1\in A$$ .

We propose an algorithm for computing the N best vertices in a weighted acyclic hypergraph over a nice semiring. A semiring is nice if it is finitely-generated, idempotent, and has 1 as its minimal element. We then apply the algorithm to the problem of computing the N best trees with respect to a weighted tree automaton, and complement theoretical correctness and complexity arguments with experimental data. The algorithm has several practical applications in natural language processing, for example, to derive the N most likely parse trees with respect to a probabilistic context-free grammar.

A subset A of a continuum X is called T -closed set if T ( A ) = A , where T denotes the Jones T - function. We give a characterization of T -closed sets and then we show a counterexample to a hypothesis by David P. Bellamy, Leobardo Fernandez and Sergio Macias about T -closed sets if T is idempotent on X.

In this paper, we study linear operators on real and complex Euclidean spaces which are real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We can compare some partial order properties of the orthogonal and of the R-orthogonal projections. We prove that the set of all R-orthogonal projections in finite-dimensional complex space is a quantum logic.

The power semigroup, or global, of a semigroup S is the set 𝒫(S) of all nonempty subsets of S equipped with the naturally defined multiplication. A class 𝒦 of semigroups is globally determined if any two members of 𝒦 with isomorphic globals are themselves isomorphic. The principal goal of this paper is to prove that the class of all idempotent semigroups is globally determined.

By weakening the neutral element condition of semiuninorms, we introduce a new concept called weak-neutral semiuninorms (shortly, wn-semiuninorms). After analyzing their structure, several classes of wn-semiuninorms are presented and discussed. Particularly, based on a kind of monotone unary functions which are not necessarily continuous and strictly monotone, we introduce representable wn-semiuninorms and discuss some of their properties in detail. We show that there is no idempotent proper wn-semiuninorm. Each representable wn-semiuninorm is Archimidean but not strictly monotone, and its additive generator is unique up to a positive multiplicative constant under some conditions. In the discussion about the representable wn-semiuninorms, we also characterize the solutions to a class of Cauchy functional equations on a restricted domain.

Several papers deal with the problem of counting the number of idempotent endomorphisms of a structure S onto a substructure T. In this paper we consider the case when T is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Suppose A is the image (=range) of an idempotent endomorphism of the free n-generator MV-algebra M ( [ 0 , 1 ] n ) of McNaughton functions on [ 0 , 1 ] n . We prove that the number r ( A ) of idempotent endomorphisms of M ( [ 0 , 1 ] n ) onto A is finite if, and only if, the maximal spectral space μ A is homeomorphic to a (Kuratowski) closed domain M of [ 0 , 1 ] n , in the sense that M = cl ( int ( M ) ) . Further, the closed domain condition is decidable and r ( A ) is computable, once an idempotent endomorphism of M ( [ 0 , 1 ] n ) onto A is explicitly given. Thus every finitely generated projective MV-algebra B comes equipped with a new invariant ι ( B ) = sup ⁡ { r ( A ) | A ≅ B , for A the image of an idempotent endomorphism of M ( [ 0 , 1 ] k ) } , and k the smallest number of generators of B. We compute ι ( B ) for many projective MV-algebras B existing in the literature. Various problems concerning idempotent endomorphisms of free MV-algebras are shown to be decidable. Via the Γ functor, our results and computations automatically transfer to finitely generated projective abelian l-groups with a distinguished strong unit.

Let R be an associative ring with unity. An element a ∈ R is said to be r-clean if a = e + r, where e is an idempotent and r is a von Neumann regular element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we investigate the conditions under which the group ring RG is r-clean. We show that if R is a ring and G is a locally finite p-group with p ∈ J(R), then the group ring RG is r-clean if and only if R is r-clean.

Suppose that P and Q are idempotents on a Hilbert space H, while Q = Q* and I is the identity operator in H. If U = P − Q is an isometry then U = U* is unitary and Q = I − P. We establish a double inequality for the infimum and the supremum of P and Q in H and P − Q. Applications of this inequality are obtained to the characterization of a trace and ideal F-pseudonorms on a W*-algebra. Let φ be a trace on the unital C*-algebra A and let tripotents P and Q belong to A. If P − Q belongs to the domain of definition of φ then φ(P − Q) is a real number. The commutativity of some operators is established.

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ℕ. We prove that M n (R) is nil clean if and only if R/J(R) is Boolean and M n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ℤ3, B or ℤ3 ⊕ B where B is a Boolean ring, and that M n (R) is weakly nil clean if and only if M n (R) is nil clean for all n ≥ 2.

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

Let ΠL1 denote a direct power of L1, the two-element left zero semigroup with identity adjoined. A semigroup S is called left quasi-ample if for each a∈S there exists a unique idempotent a+∈S such that xa=ya⇔xa+=ya+ for all x, y∈S1 and the left ample condition e2=e⇒(ae)+a=ae holds. Generalizing a recent result in [3], we prove that the semigroups in the title are embeddable into certain transformation semigroups. Our embedding provides an easy way to construct (finite) proper covers for (finite) such semigroups. Moreover, we show that each proper such semigroup is embeddable into a semidirect product of a ΠL1-embeddable band by a right cancellative monoid, giving a partial answer to a question raised in [1].

Given an MV-algebra A, with its natural partial ordering, we consider in A the intervals of the form [0, a], where \({a \in A}\). These intervals have a natural structure of MV-algebras and will be called the relative subalgebras of A (in analogy with Boolean algebras). We investigate various properties of relative subalgebras and their relations with the original MV-algebra.

Let $$\mathcal {V}$$ be the variety of square-increasing idempotent semirings. Its members can be viewed as semilattice-ordered monoids satisfying $$x\le x^{2}$$ . We show that the universal theory of $$\mathcal {V}$$ is decidable. In order to prove this result, we investigate the class $$\mathcal {Q}$$ whose members are ordered-monoid subreducts of members from $$\mathcal {V}$$ . In particular, we prove that finitely generated members from $$\mathcal {Q}$$ are well-partially-ordered and residually finite.

A ring R is called clean if every element of it is a sum of an idempotent and a unit. A ring R is neat if every proper homomorphic image of R is clean. When R is a field, then a complete ch...

Let $${\mathcal {R}}$$ be a unital prime $$*$$ -ring containing a nontrivial symmetric idempotent. For $$A,B\in {\mathcal {R}}$$ , the skew commutator and 2-skew commutator are defined, respectively, by $${}_*[A,B]=AB-BA^*$$ and $${}_*[A,B]_2= {{}_*[A, {{}_*[A,B]}]}$$ . Let $$\Phi :{\mathcal {R}} \rightarrow {\mathcal {R}}$$ be a surjective map. We show that (1) $$\Phi $$ satisfies $${}_*[\Phi (A),\Phi (B)] = {{}_*[A,B] }$$ for all $$A, B\in {\mathcal {R}}$$ if and only if there exists $$\lambda \in \{-1,1\}$$ such that $$\Phi (A)=\lambda A$$ for all $$A\in {\mathcal {R}}$$ ; (2) $$\Phi $$ satisfies $${}_*[\Phi (A),\Phi (B)]_2= {{}_*[A,B]_2}$$ for all $$A, B\in {\mathcal {R}}$$ if and only if there exists $$\lambda \in {\mathcal {C}}_S $$ with $$\lambda ^{3} = I$$ such that $$\Phi (A) = \lambda A $$ for all $$A \in {\mathcal {R}}$$ , where I is the unit of $${\mathcal {R}}$$ and $${\mathcal {C}}_S $$ is the symmetric extend centroid of $${\mathcal {R}}$$ . This is then applied to prime $$\hbox {C}^*$$ -algebras, factor von Neumann algebras and indefinite self-adjoint standard operator algebras to get a complete invariant for the identity map and to symmetric standard operator algebras as well as matrix algebras.

Let be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent. Assume that is a surjective map. It is shown that f is strong 3-commutativity preserving, that is, f satisfies for all , if and only if for all , where is an element in the extended centroid of with and is a map from into its extended centroid. Applications to prime C-algebras, factor von Neumann algebras, standard operator algebras and matrix algebras are also obtained .

We study and propose some new construction methods to obtain uninorms on bounded lattices. Considering an arbitrary bounded lattice L, we show the existence of idempotent uninorms on L for any element e ź L\{0, 1} playing the role of a neutral element. By our construction method, we obtain the smallest idempotent uninorm and the greatest idempotent uninorm with the neutral element e ź L\{0, 1}. We see that the obtained uninorms are conjunctive and disjunctive uninorms, respectively. On the other hand, if L is not a chain, we also provide an example of an idempotent uninorm which is neither conjunctive nor disjunctive.

We study bi-additive maps of ξ-Lie product type satisfying certain local property from the viewpoint of functional identities. Let R be a unital prime ring with extended centroid C, ξ∈C, and maximal left ring of quotients Qml(R). Suppose that R contains a nontrivial idempotent. We completely characterize additive maps ϕ:R→Qml(R) having the property that [ϕ(x),y]ξ+[x,ϕ(y)]ξ=0 whenever x,y∈R satisfy xy = 0 = yx. As consequences, some known results are fully generalized.

Let 𝒜 be a unital ring, let ℳ be a unital 𝒜-bimodule. Suppose that 𝒜 contains a wide idempotent (with respect to ℳ). The problem of describing the form of maps F1,F2,G1,G2:𝒜→ℳ satisfying F1(x)y+F2(y)x+xG2(y)+yG1(x)=0 for all x,y∈𝒜 is considered. As an application, generalized inner biderivations on rings with wide idempotents are determined. As another application, a short proof of a result of Bresar on range-inclusive maps is given.

A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a ∈ R, there exists an idempotent e ∈ ℤ[a] such that a − e ∈ N(R), if and only if R is periodic and R∕J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove that R is strongly nil clean if and only if for all a ∈ R, there exist n ∈ ℕ,k ≥ 0 (depending on a) such that an−an+2k∈N(R), if and only if for fixed m,n ∈ ℕ, a−a2n+2m(n+1)∈N(R) for all a ∈ R. These also extend known theorems, e.g, [5, Theorem 3.21], [6, Theorem 3], [7, Theorem 2.7] and [12, Theorem 2].

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties—torsion and almost-divisibility (an element \(a\in S\) is called almost-divisible in S if there is \(b\in \mathbb {N}\cdot a\) such that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements \(f\in F(X)\) such that for every epimorphism \(\pi \) of F(X) torsion and almost-divisibility of \(\pi (f)\) are equivalent in \(\pi (F(X))\).

The paper presents a full description of idempotent elements of the semigroup of binary relations BX(D), which are defined by semilattices of the class Σ3(X, 8). For the case where X is a finite set and Z7 = O, we derive formulas for calculating the number of idempotent elements of the respective semigroup.