A lattice L is projective in a variety V of lattices if whenever f : K L is an epimorphism, there is a homomorphism g : L→ K such that f(g(a)) = a for all a ∈ L. Note g is one-to-one and ρ = gf : K→ K is a retraction of K; that is, ρ is an endomorphism of K and ρ2 = ρ. The sublattice ρ(K) = g(L) of K is isomorphic to L. The image g(L) is a retract of K. In a slight abuse of terminology, we will...

Høholdt, van Lint and Pellikaan proposed a generalization of one-point AG codes called the evaluation codes, and enabled us to construct linear codes from arbitrary algebraic varieties [5], [6]. To construct an evaluation code, we find an appropriate K-algebra R with an order function w and an epimorphism φ : R → Kn of K-algebras, where K is a finite field and n is the code length. Then we defi...

Although originally devised to define the higher algebraic K-theory of rings [ 1, 7, 161, the plus-construction has quickly established its usefulness in such diverse areas as stable homotopy theory [ 15 J, bordism of manifolds [lo], and the study of knot complements [ 141. Recall (from, e.g., 11 ] h w ose notation we foHow) that the plusconstruction qx : X --, X+ is a pointed cofibration which...

Let G be a connected real Lie group. The universal representation kernel, Ko, oí G is defined as the intersection of all kernels of continuous finite dimensional representations of G. Evidently, Ko is a closed normal subgroup of G, and it is known from a theorem due to Goto (cf. [l, Theorem 7.1]) that G/Ko has a faithful continuous finite dimensional representation. Thus Kg is the smallest norm...

In this paper we examine on a pair of adjoint functors (ϕ^* ,ϕ_*)for subcategory the category crossed modules over commutative algebras where ϕ_*: XMod/P → XMod/Q, induced, and ϕ^*:XMod/Q XMod/P, pullback (co-induced), which enables us to move from Q-modules P-modules by an algebra morphism ϕ : P Q. We show that functor (ϕ^*,ϕ_*) makes p∶ XMod k-Alg into bi- fibred k-Alg, algebras, p is given p...

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, not injective. In particular this equivalent to saying $BS(2,3)$ non-trivial quotient isomorphic itself. As consequence the Cayley graph of itself up change generators. We describe on graph-level and take closer look at most common $\phi$. show its kernel free infinite ...

This paper deals with several aspects of epimorphisms in the category MFrm of metric frames and contractive homomorphisms. In particular, it is shown that (i) the epicomplete metric frames are uniquely determined by the power-set lattices of sets, (ii) episurjective is the same as Boolean, (iii) a metric frame has an epicompletion iff it is spatial, and (iv) the subcategory of epicomplete L in ...

This thesis provides a model–theoretic semantic analysis of aspects of the LF logical framework. The LF logical framework is the λΠ-calculus together with the judgements-as-types representation mechanism. A denotational semantics is provided for the λΠ-calculus in terms of Kripke λΠ-models. These are a generalization of the Kripke lambda models of Mitchell and Moggi to dependent types and are b...

Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ ${B}U$ have finite flat dimensions, then left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ Ding projective if only $M\_1$ $M\_2/{\operatorname{im}(\varphi^{M})}$ the morphism $...

Abstract Let G be a finitely generated group that can written as an extension $$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} \stackrel{f}{\longrightarrow} \Gamma \end{align*} where is group. By study of the Bieri–Neumann–Strebel (BNS) invariants we prove if $b_1(G)> b_1(\Gamma ) > 0$ , then algebraically fibres; is, admits epimorphism to $\Bbb {Z}$ with kernel. An int...