N Sequence Prime Ideals

In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

Variations of essentiality of ideals in commutative rings

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on [Formula: see text] to characterize spaces [Formula: see text] when every irreducible ideal of [Formula: see text] is pseudoprime. We also characterize the rings of continuous functions [Formula: see text] in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam.

Properties of Casson–Gordon’s rectangle condition

For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum.

Studying Semigroups Using the Properties of Their Prime m-Ideals

In this article, we present the idea of m-ideals, prime m-ideals and their associated types for a positive integer m in a semigroup. We present different chrarcterizations of semigroups through m-ideals. We demonstrate that the ordinary ideals, and their relevent types differ from the m-ideals and their assocated types by presenting concrete examples on the maximal, irreducible and strongly irreducible m-ideals. We conclude from the study that the introduction of the m-ideal will explore new fields of studies in semigroups and their applications.

T-fuzzy Bi-ideals in Semirings

In this work, we initiate the study of fuzzy bi-ideals under t-norms (T-fuzzy bi-ideals) in semirings and investigate some properties of them. Also we define prime, strongly prime, semiprime, irreducible, strongly irreducible T-fuzzy bi-ideals of semirings. Next we investigate them under regular and intra-regular semirings. Finally, we characterize them under totally ordered by inclusion of T-fuzzy bi-ideals of semirings.