In this paper we study the notion of Smarandache loops. We obtain some interesting results about them. The notion of Smarandache semigroups homomorphism is studied as well in this paper. Using the definition of homomorphism of Smarandache semigroups we give the classical theorem of Cayley for Smarandache semigroups. We also analyze the Smarandache loop homomorphism. We pose the problem of findi...

By constructions in monoid and group theory we exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms, and the category of partially ordered groups and group homomorphisms, such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of lazy homomorphism for a...

Let (M,ω) be a closed symplectic manifold and Ham(M,ω) the group of Hamiltonian diffeomorphisms of (M,ω). Then the Seidel homomorphism is a map from the fundamental group of Ham(M,ω) to the quantum homology ring QH∗(M ; Λ). Using this homomorphism we give a sufficient condition for when a nontrivial loop ψ in Ham(M,ω) determines a nontrivial loop ψ × idN in Ham(M ×N, ω ⊕ η), where (N, η) is a c...

ρ(e)(x) = e · x A1 = x = 1X(x) for all g, h ∈ G and x ∈ X. Thus ρ(gh) = ρ(g) ◦ ρ(h), ρ(e) = 1X and ρ is hence a homomorphism of monoids G → M(X). Then we note 1X = ρ(e) = ρ(gg ) = ρ(g) ◦ ρ(g) and similarly 1X = ρ(g) ◦ ρ(g) and so the ρ(g) are bijections G → G for all g ∈ G. Thus ρ is a homomorphism G → Σ(X). Given a homomorphism λ : G → Σ(X), in order for it to be the action homomorphism of an ...

Given a topological group G, and a finitely generated group Γ, a homomorphism π : Γ→G is locally rigid if any nearby by homomorphism π is conjugate to π by a small element of G. In 1964, Weil gave a criterion for local rigidity of a homomorphism from a finitely generated group Γ to a finite dimensional Lie group G in terms of cohomology of Γ with coefficients in the Lie algebra of G. Here we ge...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎...

If (G, *) and (H, •) are groups, then a function f : G −→ H is a homomorphism if f (x * y) = f (x) • f (y) for all x, y ∈ G. Example: Let (G, *) be an arbitrary group and H = {e}, then the function f : G −→ H such that f (x) = e for any x ∈ G is a homomorphism. In fact, f (x * y) = e = e • e = f (x) • f (y). f (x) = x for any x ∈ G is a homomorphism. In fact, f (x * y) = x * y = f (x) * f (y). ...

Although Kirby and Siebenmann [13] showed that there are manifolds which do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern [10] constructed a closed 5–manifold M 5 so that every n–manifold, with n 5, can be triangulated if and only if M 5 can be triangulated. Moreover, M 5 admits a triangulation if and only if...

LetA be a finite-dimensional commutative algebra over the complex numbers with nonzero multiplicative identity element e. Let M denote the set of homomorphisms from A onto the complex numbers. Thus we get a homomorphism from A into the algebra of complex-valued functions on M, with respect to pointwise addition and multiplication of functions, by sending an element x of A to the function on M w...