Narrow Results

Let $R$ be a ring and $e$ an idempotent of $R$, $R$ is called an $e$-symmetric ring if $abc=0$ implies $acbe=0$ for all $a,b,c\in R$. Obviously, $R$ is a symmetric ring if and only if $R$ is a $1$-symmetric ring. In this paper, we show that a ring $R$ is $e$-symmetric if and only if $e$ is left semicentral and $eRe$ is symmetric. As an application, we show that a ring $R$ is left min-abel if and only if $R$ is $e$-symmetric for each left minimal idempotent $e$ of $R$. We also introduce the definition of strongly $e$-symmetric ring and prove that $R$ is a strongly $e$-symmetric ring if and only if $e\in Z(R)$ and $eRe$ is a symmetric ring. Finally, we introduce $e$-reduced ring and study some properties of it.

Let $R$ be an associative unital ring with an endomorphism $\alpha$ and $\alpha$-derivation $\delta$. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the $\pi$-regular elements and also the von Neumann local elements of the Ore extension ring $R[x;\alpha ,\delta ]$, when the base ring $R$ is a right duo ring which is $(\alpha ,\delta )$-compatible. As an application, we completely characterize the clean elements of the Ore extension ring $R[x;\alpha ,\delta ]$, when the base ring $R$ is a right duo ring which is $(\alpha ,\delta )$-compatible.

The concept of generalized hypersubstitutions was introduced by S. Leeratanavalee and K. Denecke as a way of making precise the concepts of strong hyperidentity and M-strong hyperidentity. The set Hyp_G(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. All idempotent and regular elements in the monoid of all generalized hypersubstitutions of type τ = (2) were studied by W. Puninagool and S. Leeratanavalee. In this paper, we determine all primitive idempotent elements of this monoid and characterize the natural partial ordering on the set of all idempotent of this monoid.

A mapping σ which assigns to every n-ary cooperation symbol f_i an n_i-ary coterm of type τ = (n_i)_i∈I is said to be a cohypersubstitution of type τ. The concepts of cohypersubstitutions were introduced in [Cohyperidentities and M-solid classes of coalgebras, Discrete Math.309(4) (2009) 772–783]. Every cohypersubstitution σ of type τ induces a mapping on the set of all coterms of type τ. The set of all cohypersubstitutions of type τ under the binary operation which is defined by for all σ₁, σ₂ ∈ Cohyp(τ) forms a monoid which is called the monoid of cohypersubstitution of type τ. In this research, we characterize all idempotent and regular elements of Cohyp(τ) where τ = (3).

A mapping $\sigma$ from $\{f_i|i\in I\}$, the set of all co-operation symbols of type $\tau$, into $C{T}_{\tau }$, the set of all coterms of type $\tau$, is said to be a generalized cohypersubstitution of type $\tau$. Every generalized cohypersubstition $\sigma$ of type $\tau$ induces a mapping $\widehat{\sigma }$ on the set of all coterms of type $\tau$. The set of all generalized cohypersubstitutions of type $\tau$, ${\mathrm{\text{Cohyp}}}_G(\tau )$, under the binary operation ${\circ }_{CG}$, which is defined by ${\sigma }_1{\circ }_{CG}{\sigma }_2:={\widehat{\sigma }}_1\circ {\sigma }_2$ for all ${\sigma }_1,{\sigma }_2\in {\mathrm{\text{Cohyp}}}_G(\tau )$, forms a monoid which is called the monoid of generalized cohypersubstitution of type $\tau$. In this research, we characterize all idempotent and regular elements of ${\mathrm{\text{Cohyp}}}_G(\tau )$, where $\tau =(3)$.

A linear tree language of type $\tau$ is a set of linear terms, terms containing no multiple occurrences of the same variable, of that type. Instead of the usual generalized superposition of tree languages, we define the generalized linear superposition to deal with linear tree languages and study its properties. Using this superposition, we define the product of linear tree languages. This product is not associative on the collection of all linear tree languages, but it is associative on some subsets of this collection whose products of any element in the subsets are nonempty. We classify such subsets and study properties of the obtained semigroup especially idempotent elements, regular elements, and Green’s relations ${ℒ}$ and ${ℛ}$.

A linear tree language of type $\tau$ is a set of linear terms, terms in which each variable occurs at most once, of that type. We investigate a semigroup consisting of the collection of all linear tree languages such that products of any element in the collection are nonempty and the operation of the corresponding linear product especially idempotent elements, Green’s relations ${\mathscr{H}}$, ${𝒟}$, and ${𝒥}$, and some of its subsemigroups. We discover that this semigroup is neither factorizable nor locally factorizable. We also study the linear product iteration and show that any iteration is idempotent in this semigroup. Moreover, we study a semigroup with the complement of the universe set of the above semigroup together with the same linear product operation.

The set of all $n$-ary terms of type $\tau$ together with a binary operation derived from a superposition $S^n$ forms various forms of semigroups. One may generalize such binary operation by deriving it from an inductive composition of terms and call it an inductive product. However, this operation is not associative on the same base set but it becomes associative when all elements of subterms of a fixed term used in an inductive product except itself are excluded from the base set. Hence, a semigroup is formed. In this paper, we mainly focus on the algebraic structures of this semigroup such as idempotent elements, elements associating with each type of regularity condition, and Green’s relations. The formulae of complexity of inducted terms are also under investigation.