Let $R$ be a commutative ring with identity and $\mathcal{R}(_{R}M)$ denotes the bounded lattice of radical submodules of an $R$-module $M$. We say that $M$ is a radical distributive module, if $\mathcal{R}(_{R}M)$ is a distributive lattice. It is shown that the class of radical distributive modules contains the classes of multiplication modules and finitely generated distributive modules properly. It is shown that if $M$ is a semisimple $R$-module and for any radical submodule $N$ of $M$ with direct sum complement $\tilde{N}$, the complementary operation on $\mathcal{R}(_{R}M)$ is defined by $N':=\tilde{N}+rad(0)$, then $\mathcal{R}(_{R}M)$ with this unary operation forms a Boolean algebra. In particular, if $M$ is a multiplication module over a semisimple ring $R$, then $\mathcal{R}(_{R}M)$ is a Boolean algebra, which is also a homomorphic image of $\mathcal{R}(_{R}R)$.

The aim of this paper is to construct an algebraic hyperstructure over a set G corresponding to a Boolean algebra B and a function S:GB. In order to accomplish this goal we will need to define a hyperoperation on the set G. We define, ab =a^6 b={gEG/s(g)<=s(a)vs(b)} and prove that if the image of G is a v -semilattice or constitute a partition of 1 in B, then (G, ^6) is a hypergroup.

For a commutative semiring S, by an S-algebra we mean a commutative semiring A equipped with a homomorphism S → A. We show that the subvariety of S-algebras determined by the identities 1 + 2x = 1 and x 2 = x is closed under non-empty colimits. The (known) closedness of the category of Boolean rings and of the category of distributive lattices under non-empty colimits in the category of commutative semirings both follow from this general statement.

Let X and Y be vector spaces. We show that a mapping f : X ®Y satisfies the functional equation, f  (x1 +S2dj=2(-1)j xj)-f (x1+S2dj=2(-1)j-1xj)=2S2dj=2 (-1)j (xj)if and only if the mapping f : X ® Y is Cauchy additive, and prove the Cauchy-Rassias stability of the above functional equation in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let A and B be unital C*-algebras, Poisson C*-algebras or Poisson JC*-algebras. As an application, we show that every almost homomorphism h  A®B of A into B is a homomorphism when h(2nuy) = h(2nu)h(y) or h(2nu°y) = h(2nu) o h(y), for all unitaries u Î A, all y Î A, and n = 0, 1, 2, Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras or Poisson JC*-algebras. 

In this paper, some new properties of EQ-algebras are investigated. We intro-duce and study the notion of Boolean center of lattice ordered EQ-algebras with bottom element. We show that in a good ℓ EQ-algebra E with bottom element the complement of an element is unique. Furthermore, Boolean elements of a good bounded lattice EQ-algebra are characterized. Finally, we obtain conditions under which Boolean center of an EQ-algebra E is the subalgebra of E.

This paper presents a new method for simplification of Boolean functions based on Boolean differences. The proposed method is applicable to various forms of Boolean functions, including truth tables and Binary Decision Diagrams (BDDs). The Boolean differences are extended to cover the truth tables with don't-care components and cutset graphs in BDDs. The results of simplification agree with Quine-McCluskey and ESPRESSO methods. Experimental tests on MCNC and Berkeley PLA benchmarks show that the proposed method gains a performance of 1.5-10 times faster than ESPRESSO. The algorithms of the proposed method are implemented in Java/Perl/C++, and a toolset for logic function simplification is developed.