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Register a new userThe Semiring Properties of Boolean Propositional Algebras
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This paper illustrates the relationship between boolean propositional algebra and semirings, presenting some results of partial ordering on boolean propositional algebras, and the necessary conditions to represent a boolean propositional subalgebra as equivalent to a corresponding boolean propositional algebra. It is also shown that the images of a homomorphic function on a boolean propositional algebra have the relationship of boolean propositional algebra and its subalgebra. The necessary and sufficient conditions for that homomorphic function to be onto-order preserving, and also an extension of boolean propositional algebra, are explored.
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The relationship between Heyting algebras (HA) and semirings is explored. A new class of HAs called Symmetric Heyting algebras (SHAs) is proposed, and a necessary condition on SHAs to be consider semirings is given. We define a new mathematical family called Heyting structures, which are similar to semirings, but with Heyting-algebra operators in place of the usual arithmetic operators usually seen in semirings. The impact of the zero-sum free property of semirings on Heyting structures is shown as also the condition under which it is possible to extend one Heyting structure to another. It is also shown that the union of two or more sets forming Heyting structures is again a Heyting structure, if the operators on the new structure are suitably derived from those of the component structures. The analysis also provides a sufficient condition such that the larger Heyting structure satisfying a monotony law implies that the ones forming the union do so as well.
We provide a constructive treatment of basic results in the theory of central simple algebras. One main issue is the fact that one starting result, Wedderburns Theorem stating that a simple algebra is a matrix algebra over a skew field, is not constructively valid. We solve this problem by proving instead a dynamical version of this theorem. One can use this to give constructive proofs of basic results of the theory of central simple algebras, such as Skolem-Noether Theorem. We illustrate this development by giving an elementary constructive proof of a theorem of Becher (which is itself a consequence of a celebrated theorem of Merkurjev).
We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective etale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section.
Fix a pair of relatively prime integers $n>kge 1$, and a point $(eta , | , tau) in mathbb{C} times mathbb{H}$, where $mathbb{H}$ denotes the upper-half complex plane, and let ${{a ; ,b} choose {c , ; d}} in mathrm{SL}(2,mathbb{Z})$. We show that Feigin and Odesskiis elliptic algebras $Q_{n,k}(eta , | , tau)$ have the property $Q_{n,k} big( frac{eta}{ctau+d} ,bigvert , frac{atau+b}{ctau+d} big) cong Q_{n,k}(eta , | , tau)$. As a consequence, given a pair $(E,xi)$ consisting of a complex elliptic curve $E$ and a point $xi in E$, one may unambiguously define $Q_{n,k}(E,xi):=Q_{n,k}(eta , | , tau)$ where $tau in mathbb{H}$ is any point such that $mathbb{C}/mathbb{Z}+mathbb{Z}tau cong E$ and $eta in mathbb{C}$ is any point whose image in $E$ is $xi$. This justifies Feigin and Odesskiis notation $Q_{n,k}(E,xi)$ for their algebras.
Let $R$ be a commutative additively idempotent semiring. In this paper, some properties and characterizations for permanents of matrices over $R$ are established, and several inequalities for permanents are given. Also, the adjiont matrices of matriecs over $R$ are considered. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices.
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