EQUATIONS OVER DIRECT POWERS OF ALGEBRAIC STRUCTURES IN RELATIONAL LANGUAGES

For a semigroup S (group G) we study relational equations and describe all semigroups S with equationally Noetherian direct powers. It follows that any group G has equationally Noetherian direct powers if we consider G as an algebraic structure of a certain relational language. Further we specify the results as follows: if a direct power of a finite semigroup S is equationally Noetherian, then the minimal ideal Ker(S) of S is a rectangular band of groups and Ker(S) coincides with the set of all reducible elements

A Case Study of Holomorphic Semigroups

In this paper, we investigate some characteristic features of holomorphic semigroups. In particular, we investigate nice examples of holomorphic semigroups whose every left or right ideal includes minimal ideal. These examples are compact topological holomorphic semigroups.

Varieties of ∗-regular rings

Given a subdirectly irreducible ∗-regular ring R, we show that R is a homomorphic image of a regular ∗-subring of an ultraproduct of the (simple) eRe, e in the minimal ideal of R; moreover, R (with unit) is directly finite if all eRe are unit-regular. For any subdirect product of artinian ∗-regular rings we construct a unit-regular and ∗-clean extension within its variety.

KONDISI MINIMAL IDEAL KIRI TERURUT PADA SEMIGRUP TERNER TERURUT PARSIAL

Ternary semigroups 𝑇 is obtained from a nonempty set 𝑇 that given a mapping with a multiplication operation ternary that satisfied closed and associative properties. So, generally a ternary semigroup is an abstraction of a semigroup structure. Meanwhile, partially ordered ternary semigroups 𝑇 is an ordered semigroup 𝑇 that satisfies the properties for each 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑇 if 𝑎 ≤ 𝑏 then (𝑎𝑐𝑑) ≤ (𝑏𝑐𝑑) and (𝑑𝑐𝑎) ≤ (𝑑𝑐𝑏). In a ternary semigroups there is also concept of left ideals. This study was conducted to examine the characteristics of ordered left ideals on partially ordered ternary semigroups. Furthermore, it will be discussed about the characteristics of minimal ordered left ideals on partially ordered semigroups.Keywords : Ternary Semigroups, Ordered Ternary Semigroups, Left Ideals, Ordered Left Ideals, Minimal of Ordered Left Ideals.

Random shuffles on trees using extended promotion

The Tsetlin library is a very well-studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. In this paper, we consider a generalization which can be interpreted as a self-organizing library in which the arrangements of books on each shelf are restricted to be linear extensions of a fixed poset. The moves on the books are given by the extended promotion operators of Ayyer, Klee and Schilling while the shelves, bookcases, etc. evolve according to the move-to-back moves as in the the self-organizing library of Björner. We show that the eigenvalues of the transition matrix of this Markov chain are [Formula: see text] integer combinations of the transition probabilities if the posets that prescribe the restrictions on the book arrangements are rooted forests or more generally, if they consist of ordinal sums of a rooted forest and so called ladders. For some of the results, we show that the monoids generated by the moves are either [Formula: see text]-trivial or, more generally, in [Formula: see text] and then we use the theory of left random walks on the minimal ideal of such monoids to find the eigenvalues. Moreover, in order to give a combinatorial description of the eigenvalues in the more general case, we relate the eigenvalues when the restrictions on the book arrangements change only by allowing for one additional transposition of two fixed books.

On structure of the semigroups of k-linked upfamilies on groups

Given a group [Formula: see text], we study right and left zeros, idempotents, the minimal ideal, left cancelable and right cancelable elements of the semigroup [Formula: see text] of [Formula: see text]-linked upfamilies and characterize groups [Formula: see text] whose extensions [Formula: see text] are commutative. We finish the paper with the complete description of the structure of the semigroups [Formula: see text] for all groups [Formula: see text] of cardinality [Formula: see text].

Superextensions of cyclic semigroups

Given a cyclic semigroup $S$ we study right and left zeros, singleton left ideals, the minimal ideal, left cancelable and right cancelable elements of superextensions $\lambda(S)$ and characterize cyclic semigroups whose superextensions are commutative.

ON SUPERNILPOTENT NONSPECIAL RADICALS

Let ρ be a supernilpotent radical. Let ρ* be the class of all rings A such that either A is a simple ring in ρ or the factor ring A/I is in ρ for every nonzero ideal I of A and every minimal ideal M of A is in ρ. Let $\mathcal {L}\left ( \rho ^{\ast }\right ) $ be the lower radical determined by ρ* and let ρφ denote the upper radical determined by the class of all subdirectly irreducible rings with ρ-semisimple hearts. Le Roux and Heyman proved that $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a supernilpotent radical with $\rho \subseteq \mathcal {L}\left ( \rho ^{\ast }\right ) \subseteq \rho _{\varphi }$ and they asked whether $\mathcal {L} \left ( \rho ^{\ast }\right ) =\rho _{\varphi }$ if ρ is replaced by β, ℒ , 𝒩 or 𝒥 , where β, ℒ , 𝒩 and 𝒥 denote the Baer, the Levitzki, the Koethe and the Jacobson radical, respectively. In the present paper we will give a negative answer to this question by showing that if ρ is a supernilpotent radical whose semisimple class contains a nonzero nonsimple * -ring without minimal ideals, then $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a nonspecial radical and consequently $\mathcal {L}\left ( \rho ^{\ast }\right ) \neq \rho _{\varphi }$. We recall that a prime ring A is a * -ring if A/I is in β for every $0\neq I\vartriangleleft A$.