Implications in a category can be presented as epimorphisms: an object satisfies the implication iff it is injective w.r.t. that epimorphism. G. Roçu formulated a logic for deriving an implication from other implications. We present two versions of implicational logics: a general one and a finitary one (for epimorphisms with finitely presentable domains and codomains). In categories Alg Σ of al...

Given a compact, connected, oriented 3-manifold M with boundary, and epimorphism χ from H1M to a free abelian group Π, two invariants β, τ ∈ ZΠ are defined. If M embeds in another such 3manifold N such that χN factors through χ, then the product βτ divides ∆0(H1Ñ). A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein theoretic generalizatio...

Let K be the kernel of an epimorphism χ : G → Z, for G a finitely presented group. If K has uncountably many normal subgroups of finite index r, then K has uncountably many subgroups (not necessarily normal) of any finite index greater than r. In particular, this is the case whenever G is subgroup separable and K is nonfinitely generated. Assume that G has an abelian HNN base contained in K. If...

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → LX is an epimorphism for all X and the class of L-local objects is defined by an absolu...

A probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities if the morphism is either a...

As a sequel of [4], this paper is devoted to the computation of the digital fundamental group πd 1(O/S;σ) defined by loops in the digital object O for which the digital object S acts as an “obstacle”. We prove that for arbitrary digital spaces the group πd 1(O/S;σ) maps onto the usual fundamental group of the difference of continuous analogues |AO∪S | − |AS |. Moreover, we show that this epimor...

We first give a short group theoretic proof of the following result of Lackenby. If G is a large group, H is a finite index subgroup of G admitting an epimorphism onto a non–cyclic free group, and g is an element of H, then the quotient of G by the normal subgroup generated by g n is large for all but finitely many n ∈ Z. In the second part of this note we use similar methods to show that for e...

We say that an algebra is zero-product balanced if ab⊗c and a⊗bc agree modulo tensors of elements with zero-product. This closely related to but more general than the notion a determined introduced developed by Brešar, Villena others. Every surjective, preserving map from automatically weighted epimorphism, this implies algebras are their linear structure. Further, commutator subspace can be de...

Let R be a ring. A right R-module is said to C-flat if the kernel of any epimorphism B → C-pure in B, i.e. induced map Hom(C,B) Hom(C,A) surjective for cyclic C. Projective modules are and weakly-flat neat-flat. In this article, it discussed connections between C-flat, weakly-flat, neat-flat singly flat modules. It shown that coincide with singly-projective over arbitrary rings. Next, several c...

Consider a group G and an epimorphism u0 : G → Z inducing a splitting of G as a semidirect product ker(u0)oφ Z with ker(u0) a finitely generated free group and φ ∈ Out(ker(u0)) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized G as π1(X) for an Eilenberg-Maclane 2–complex X equipped with a semiflow ψ, and inspired by McMullen’s T...