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.
In the present paper we generalize the notion of a Heyting algebra to the non-commutative setting and hence introduce what we believe to be the proper notion of the implication in skew lattices. We list several examples of skew Heyting algebras, including Heyting algebras, dual skew Boolean algebras, conormal skew chains and algebras of partial maps with poset domains.
In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. We present several results addressed toward a semiring theory for MV-algebras. In particular we show a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, the construction of the Grothendieck group of a semiring and its functorial nature, and the effect of Mundici categorical eq…
We give a new order-theoretic characterization of a complete Heyting and co-Heyting algebra $C$. This result provides an unexpected relationship with the field of Nash equilibria, being based on the so-called Veinott ordering relation on subcomplete sublattices of $C$, which is crucially used in Topkis' theorem for studying the order-theoretic stucture of Nash equilibria of supermodular games.
Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras
Acharya Nagarjuna University（阿查里亚·纳加尔朱纳大学）；Government College (Autonomous), Rajahmundry（拉贾穆恩德里政府学院（自治））；Government College for Women (Autonomous), Guntur（贡图尔女子政府学院（自治））
Here we characterize the linear operators that preserve rank of matrices over additively idempotent and multiplicatively cancellative semirings. The main results in this article generalize the corresponding results on the two element Boolean algebra and on the max algebra; and holds on max-plus algebra and some other tropical semirings.
We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all $n \times n$ upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. In the $2 \times 2$ case this also yields a representation as a subsemigroup of a semidirect product of commutative monoids. In particular, from the case where $n=2$ and the semiring is the tropical semifield, we obtain a representation of the free objects in the monoid and semigroup varieties gener…
We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the well-known characterizations when S is a field or a commutative ring. We also provide a criterion for the Steinberg algebra of the graph groupoid associated to an arbitrary graph to be congruence-simple. Motivated by a result of Clark and Sims, we show that, over the Boolean semifield, the natural homomorphism from the Leavitt path algebra to the Steinberg algebr…
In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $SK_0(S)$ of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the $SK_0$-groups and $K_0$-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the $SK_0$-groups completely characterize zerosumfree congruence-semisimple semirings.
In this paper we develop a number of results and notions concerning Positivstellensätze for semirings (preprimes) of commutative unital real algebras. First we reduce the Archimedean Positivstellensatz for semirings to the corresponding result for quadratic modules. Various applications of the Archimedean Positivstellensatz for semirings are investigated. A general Positivstellensatz with denominators is proved for filtered algebras with semirings. As an application we derive a denominator-free Positivstellensatz for the cylindrical extension of an algebra with Archimedean semiring. A large nu…
Di Nola and Gerla showed that MV-algebras and coupled semirings are in a natural one-to-one correspondence. We generalize this correspondence to residuated lattices satisfying the double negation law.
Finite Structure and Radical Theory of Commutative Ternary $Γ$-Semirings
Department of Mathematics, Government College (Autonomous), Y-Junction, Rajahmundry, 533105, A.P.,India（政府学院（自治），Y交汇处，拉贾穆恩德里）；Department of Mathematics, Acharya Nagarjuna University, Pedakakani, Guntur, 522510, A.P., India（阿查里亚·纳加尔琼纳大学）；Department of Mathematics, Government College For Women (A), Pattabhipuram, Guntur, 522006, A.P., India（女子政府学院（A））
We prove an open mapping theorem for the topological spaces dual to finitely presented Heyting algebras. This yields in particular a short, self-contained semantic proof of the uniform interpolation theorem for intuitionistic propositional logic, first proved by Pitts in 1992. Our proof is based on the methods of Ghilardi & Zawadowski. However, our proof does not require sheaves nor games, only basic duality theory for Heyting algebras.
Exact interval solutions to the discrete Bellman equation and polynomial complexity of problems in interval idempotent linear algebra
本文构造了偏序半环（包括非负实数半环和所有幂等半环）上矩阵区间线性方程X=AX+B（离散平稳Bellman方程）的解.本文还讨论了区间幂等线性代数问题的计算复杂性。在传统的区间分析中，这类问题一般是NP难的。本文考虑正半环上的矩阵方程，在这种情况下，问题的计算复杂性是多项式的。在最优化问题中，幂等元和其它正半环自然地出现。这些问题中的许多原来是线性适当的幂等半环。在这种情况下，方程组X=AX+B似乎是域上传统线性代数中通常线性系的自然模拟。B. A. Carre指出，许多著名的离散最优化算法类似于传统计算线性代数的标准算法。
L. P. Belluce, A. Di Nola and B. Gerla established a connection between MV-algebras and (dually) lattice ordered semirings by means of so-called coupled semirings. A similar connection was found for basic algebras and semilattice ordered right near semirings by the authors. The aim of this paper is to derive an analogous connection for orthomodular lattices and certain semilattice ordered near semirings via so-called coupled right orthosemirings.
Ternary Gamma Semirings as a Novel Algebraic Framework for Learnable Symbolic Reasoning
Government College (A), Rajahmundry, A.P., India（印度安得拉邦拉贾姆ун迪政府学院）；Department of Mathematics, Acharya Nagarjuna University, Guntur, A.P., India（安得拉邦古尔塔阿查里亚纳加普大学数学系）；Government College For Women(A), Guntur, Andhra Pradesh, India（印度安得拉邦古尔塔政府女子学院）；Dept. of Mathematics,Acharya Nagarjuna University, Guntur, A.P., India（安得拉邦古尔塔阿查里亚纳加普大学数学系）
From subtractive ideals of semirings to deductive and inductive sets in general algebras
National Institute for Theoretical and Computational Sciences（国家理论与计算科学研究所）；Department of Mathematics and Applied Mathematics, University of Johannesburg（约翰内斯堡大学数学与应用数学系）；Mathematics Division, Department of Mathematical Sciences, Stellenbosch University（斯坦陵布什大学数学科学系数学分部）；Department of Computing, Mathematical, and Statistical Sciences, University of Namibia（纳米比亚大学计算、数学与统计科学系）
We show that, if S is a finite semiring, then the free profinite S-semimodule on a Boolean Stone space X is isomorphic to the algebra of all S-valued measures on X, which are finitely additive maps from the Boolean algebra of clopens of X to S. These algebras naturally appear in the logic approach to formal languages as well as in idempotent analysis. Whenever S is a (pro)finite idempotent semiring, the S-valued measures are all given uniquely by continuous density functions. This generalises the classical representation of the Vietoris hyperspace of a Boolean Stone space in terms of continuou…
In this paper, we give semiring version of some classical results in commutative algebra related to Euclidean rings, PIDs, UFDs, G-domains, and GCD and integrally closed domains.
