In this paper we study the algebra L(Σ) generated by links in the manifold Σ¬[0, 1] where Σ is an oriented surface. This algebra has a filtration and the associated graded algebra L Gr (Σ) is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra ch (Σ) of chord diagrams on Σ to L Gr (Σ). We show that multiplication in L (Σ) provides a geometric way to define a de...

The Central Sets Theorem was introduced by H. Furstenberg and then afterwards several mathematicians have provided various versions extensions of this theorem. All these theorems deal with central sets, its origin from the algebra Stone–Čech compactification arbitrary semigroup, say βS. It can be proved that every closed subsemigroup βS is generated a filter. We will show that, under some restr...

The purpose of this paper is to prove that we can construct all finite dimensional irreducible nilpotent modules of type 1 inductively by using Schnizer homomorphisms for quantum algebra at roots of unity of type An, Bn, Cn, Dn or G2.

3 Abstract Algebra 32 3.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 32 3.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 32 3.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 32 3.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 The General Associative Law . . . . . . . . . . . . . . . . . . 34 3.6 Identity el...

4 Abstract Algebra 56 4.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 56 4.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 56 4.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 56 4.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 The General Associative Law . . . . . . . . . . . . . . . . . . 58 4.6 Identity el...

3 Abstract Algebra 33 3.1 Binary Operations on Sets . . . . . . . . . . . . . . . . . . . . 33 3.2 Commutative Binary Operations . . . . . . . . . . . . . . . . 33 3.3 Associative Binary Operations . . . . . . . . . . . . . . . . . . 33 3.4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.5 The General Associative Law . . . . . . . . . . . . . . . . . . 35 3.6 Identity el...

We characterize trees as median algebras and semilattices by relaxing conservativeness. Moreover, we describe median homomorphisms between products of median algebras and show that Arrow type impossibility theorems for mappings from a product A1 ˆ ̈ ̈ ̈ ˆ An of median algebras to a median algebra B are possible if and only if B is a tree, when thought of as an ordered structure.

We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.