We propose a method for computing the cohomology ring of three–dimensional (3D) digital binary–valued pictures. We obtain the cohomology ring of a 3D digital binary–valued picture I, via a simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. The usefulness of a simplicial description of the “digital” cohomology ring of 3D digital binary– valued pict...

We propose a method for computing the Z2–cohomology ring of a simplicial complex uniquely associated with a three–dimensional digital binary–valued picture I. Binary digital pictures are represented on the standard grid Z, in which all grid points have integer coordinates. Considering a particular 14–neighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically ...

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...

Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

let $a$ be an abelian topological group and $b$ a trivial topological $a$-module. in this paper we define the second bilinear cohomology with a trivial coefficient. we show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

let $s$ be an inverse semigroup and let $e$ be its subsemigroup of idempotents. in this paper we define the $n$-th module cohomology group of banach algebras and show that the first module cohomology group $hh^1_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is zero, for every odd $ninmathbb{n}$. next, for a clifford semigroup $s$ we show that $hh^2_{ell^1(e)}(ell^1(s),ell^1(s)^{(n)})$ is a banach space,...