Narrow Results

$({\phi }_{+},{\phi }_{-})$-derivations defined from a Banach–Jordan pair $V=(V^{+},V^{-})$ into a semisimple Banach–Jordan pair $W=(W^{+},W^{-})$ are automatically continuous provided that $\phi =({\phi }_{+},{\phi }_{-})$$({\phi }_{\sigma }:V^{\sigma }\to W^{\sigma })$ is an epimorphism.

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $𝔟(p,q)$ is minimal where $q\le 6$ or $p\le 100$.

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K₁ we characterize all other 2-bridge knots K₂ such that {K₁, K₂} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.

A partial order on prime knots can be defined by declaring $J\ge K$, if there exists an epimorphism from the knot group of $J$ onto the knot group of $K$. Suppose that $J$ is a 2-bridge knot that is strictly greater than $m$ distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of $J$ in terms of $m$. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number $\text{EK}(n)$ which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number $n$. We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K\ge J$ if there exists an epimorphism $f:{\pi }_1(S^3-K)\to {\pi }_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K>J$, then $\gamma (K)\ge 3\gamma (J)-4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K>J$, then $g(K)\ge 3g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average braid index of all two-bridge knots with a given crossing number.

In this paper, based on a characterization of epimorphisms of R-algebras given by Roby [15], we bring an algorithm testing whether a given finitely generated morphism f : A → B, where A and B are finitely presented affine algebras over the same Nœtherian commutative ring R, is an epimorphism of R-algebras or not. We require two computational conditions on R, which we call a computational ring.

The aim of this paper is to show that certain subsets, which are defined by commutativity conditions involving derivations, generalized derivations and epimorphisms, coincide with the center in prime or semiprime rings.

The concept of σ-ideals is introduced in almost distributive lattices (ADLs). Generalized stone ADLs are characterized in terms of their σ-ideals and α-ideals. Normal ADLs are also characterized in terms of their O-ideals and σ-ideals. Finally, a discussion is made about the epimorphic images and inverse images of σ-ideals.

We show that the special semigroup amalgam with core as a quasi-unitary subsemigroup is embedded in a semigroup. We also show that the special semigroup amalgam within the class of left [right] quasinormal bands is embeddable in a left [right] quasinormal band.

Firstly, we show that the variety of all left(right) regular bands and the variety of all normal bands are closed in the variety of all left(right) semiregular bands and the variety of all medial semigroups, respectively. Then, we show that the class of all regular medial semigroups satisfying certain condition is absolutely closed.

In this paper, we prove that the dominion of any full orthodox subsemigroup of a medial orthodox semigroup is described by the Isbell zigzag theorem in the category of medial orthodox semigroups. As a consequence, the dominions of any full completely regular subsemigroup of a medial completely regular semigroup as well as that of any full Clifford subsemigroup of a medial Clifford semigroup are also described by the Isbell zigzag theorem in the categories of all medial completely regular semigroups and of all medial Clifford semigroups, respectively.

By using Markoff (trace) maps, we give an alternative proof to a main result of the author's joint paper with Tomotada Ohtsuki and Robert Riley [8], which gives a systematic construction of epimorphisms between 2-bridge link groups.