If n≡0,1mod4, we prove a sum formula Vθ0(a0,aRn)=n⋅Vθ0(a0,aR) for the generalized Vaserstein symbol whenever R is smooth affine algebra over perfect field k with char(k)≠2 such that −1∈k×2. This enables us to generalize result of Fasel-Rao-Swan on transformations unimodular rows via elementary matrices normal algebras dimension d≥4 algebraically closed fields characteristic ≠2. As consequence, ...

Let (S,+) be an infinite commutative semigroup with identity 0. Let u, v ∈ N and let A be a u×v matrix with nonnegative integer entries. If S is cancellative, let the entries of A come from Z. Then A is image partition regular over S (IPR/S) iff whenever S \{0} is finitely colored, there exists ~x ∈ (S \{0}) such that the entries of A~x are monochromatic. The matrix A is centrally image partiti...

An atomic monoid $M$ is called length-factorial if for every non-invertible element $x \in M$, no two distinct factorizations of $x$ into irreducibles have the same length (i.e., number irreducible factors, counting repetitions). The notion length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under term 'other-half-factoriality': they used to provide a characterization uniqu...

semi-regular locales are extensions of the classical semiregularspaces. we investigate the conditions such that semi-regularizationis a functor. we also investigate the conditions such that semi-regularizationis a reflection or coreflection.

Let S be a (discrete) semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second dual of ` (S) is the Banach ...

We prove that left cancellative right hereditary monoids satisfying the dedekind height property are precisely the Zappa-Szép products of free monoids and groups. The ‘fundamental’ monoids of this type are in bijective correspondence with faithful self-similar group actions. 2000 AMS Subject Classification: 20M10, 20M50. 1 A class of left cancellative monoids This paper develops some ideas that...

Let S be a semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second dual of ` (S) is the Banach algebraM(βS...

We will examine totalising a partial operation in general algebra by using an absorbtive element, bottom, such as error flag. then focus on the simplest example of operation, namely subtraction natural numbers: n - m is undefined whenever < m. use bottom algebraic structures for numbers, especially semigroups and semirings. axiomatise this totalisation process introduce concept team, being a...