Definition 1.2. (Semigroup) A binary algebra (S;⊛) is called a semigroup if it satisfies the associativity property, i.e., (α ⊛ β)⊛ γ = α⊛ (β ⊛ γ) ∀α, β, γ ∈ S. A semigroup is called commutative if α⊛ β = β ⊛ α ∀α, β ∈ S. Definition 1.3. (Identity Element) An element 1l ∈ S is said to be a left identity of the binary algebra (S;⊛) if 1l ⊛ α = α ∀α ∈ S. Similarly, an element 1r ∈ S is said to be...

in this paper, a finite element model is developed for the fully hydro-mechanical analysis of hydraulic fracturing in partially saturated porous media. the model is derived from the framework of generalized biot theory. the fracture propagation is governed by a cohesive fracture model. the flow within the fracture zone is modeled by the lubrication equation. the displacement of solid phase, and...

Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${rm T}(G (£))$. It is the graph with all elements of £ as vertices, and for distinct $x, y in £$, the vertices $x$ and $y$ are adjacent if and only if $x vee y in {rm Z}(£)$. The basic properties of the ...

We show that in a symmetric m-convex algebra without algebraic zero divisors, any self-adjoint and invertible element is either positive or negative. As a consequence we obtain that a symmetric m-convex algebra containing no algebraic zero divisors and for which every positive element has a positive square root is isomorphic to C + Rad(A).

Let us note that there exists a non empty zero structure which is non trivial. Let us observe that every zero structure which is non trivial is also non empty. Let us mention that there exists a non trivial double loop structure which is Abelian, left zeroed, right zeroed, add-associative, right complementable, unital, associative, commutative, distributive, and integral domain-like. Let R be a...

We show that in a symmetric m-convex algebra without algebraic zero divisors, any self-adjoint and invertible element is either positive or negative. As a consequence we obtain that a symmetric m-convex algebra containing no algebraic zero divisors and for which every positive element has a positive square root is isomorphic to C + Rad(A).

Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 subsets of 2 , under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of Pw. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of PM .