This note shows how iteration of the standard process of adjoining identities and zeros to semigroups gives rise naturally to the lexicographical ordering on the additive semigroups of n-tuples of nonnegative integers and n-tuples of integers. 1. Semigroups with identities and zeros A binary operation ∗ on a set S is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S. A semigroup is a...

Let R be an Archimedean partially ordered ring in which the square of every element is positive, and N(R) the set of all nilpotent elements of R. It is shown that N(R) is the unique nil radical of R, and that N(R) is locally nilpotent and even nilpotent with exponent at most 3 when R is 2-torsion-free. R is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator...

We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of hemirelatively nonexpansive mappings and the set of solutions of an equilibrium problem and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space. Using this theorem, we obtain three new resul...

1.1 Preliminaries A code of length n over the finite field Fq is a subset of V = F n q , and it is linear if it is an Fq-subspace of V . When q = 2 we talk of binary codes and binary linear codes. The weight w(a) of an element a ∈ V is the number of non-zero entries in a. The minimum weight of a linear code C is the smallest number which is a weight of a non-zero element of C. The weight enumer...

A new class of rings are introduced which every element in the ring sum involution and tripotent elements. This called t-clean is a generalization invo-clean subclass clean rings. Some properties this investigate. For an application graph theory, defined set vertices order pairs them element. The two adjacent if only elements zero or product zero. graphs connecting, has diameter one girth three.

1.1. Field extensions. Suppose K is a field containing a subfield F . An element α ∈ K is called algebraic over F if there exists a non-zero polynomial f ∈ F [x] such that f(α) = 0. If α is algebraic over F , the set Iα(F ) := {f ∈ F [x] | f(α) = 0} is a non-zero ideal of F [x] (this is easy, but do check it). Since F [x] is a PID (it is in fact a Euclidean ring with respect to the degree funct...

Common meadows are fields expanded with a total inverse function. Division by zero produces an additional value denoted with a that propagates through all operations of the meadow signature (this additional value can be interpreted as an error element). We provide a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows of characteristic zero tha...

( ) [ ] [ ] ( ) 1 2 0 0 1 0 0 1 2 0 1 1 0 2 n m n m a b a b a b x a b a b a b x a b x + = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + + ⋅ . The coefficient operations are performed using the operations for the field from which the coefficients were taken. • Such a collection of polynomials forms a commutative ring with identity. • Nonzero field elements are considered to be zero-degree polynomials. The zero elemen...

1.1. Field extensions. Suppose K is a field containing a subfield F . An element α ∈ K is called algebraic over F if there exists a non-zero polynomial f ∈ F [x] such that f(α) = 0. If α is algebraic over F , the set Iα(F ) := {f ∈ F [x] | f(α) = 0} is a non-zero ideal of F [x] (this is easy, but do check it). Since F [x] is a PID (it is in fact a Euclidean ring with respect to the degree funct...

We study finite element methods for the displacement obstacle problem of clamped Kirchhoff plates. A unified convergence analysis is provided for C1 finite element methods, classical nonconforming finite element methods and C0 interior penalty methods. Under the condition that the obstacles are sufficiently smooth and that they are separated from each other and the zero displacement boundary co...