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Journal articles on the topic 'Semiring and lattices'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

Sharma, Tilak Raj, and Rajesh Kumar. "Lattice Ordered G􀀀Semirings." Indian Journal Of Science And Technology 17, no. 12 (2024): 1143–47. http://dx.doi.org/10.17485/ijst/v17i12.3184.

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Objectives: The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. Methods: To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. Findings: First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice ordered semirings and finally lattice ideals and morphisms. The unique feature of this

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2

Chen, Yizhi, and Xianzhong Zhao. "On Decompositions of Matrices over Distributive Lattices." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/202075.

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LetLbe a distributive lattice andMn,q(L)(Mn(L), resp.) the semigroup (semiring, resp.) ofn×q(n×n, resp.) matrices overL. In this paper, we show that if there is a subdirect embedding from distributive latticeLto the direct product∏i=1m‍Liof distributive latticesL1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroupMn,q(L)(semiringMn(L), resp.) to semigroup∏i=1m‍Mn,q(Li)(semiring∏i=1m‍Mn(Li), resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalize

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3

Tilak, Raj Sharma, and Kumar Rajesh. "Lattice Ordered G􀀀Semirings." Indian Journal of Science and Technology 17, no. 12 (2024): 1143–47. https://doi.org/10.17485/IJST/v17i12.3184.

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Abstract <strong>Objectives:</strong> The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. <strong>Methods:</strong> To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. <strong>Findings:</strong> First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice o

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4

Chajda, Ivan, and Helmut Länger. "When does a semiring become a residuated lattice?" Asian-European Journal of Mathematics 09, no. 04 (2016): 1650088. http://dx.doi.org/10.1142/s1793557116500881.

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It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is o

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5

Shao, Yong, and Xianzhong Zhao. "Distributive Lattices of M-Rectangular Divided-semirings." Algebra Colloquium 20, no. 02 (2013): 243–50. http://dx.doi.org/10.1142/s1005386713000217.

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In this paper, we first introduce the so-called M-rectangular divided-semirings and distributive lattices of M-rectangular divided-semirings. We then discuss the relations between such a semiring and its multiplicative semigroup. Finally, we investigate subdirect product decompositions of these semirings and obtain some interesting results.

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6

Nasehpour, Peyman. "On the content of polynomials over semirings and its applications." Journal of Algebra and Its Applications 15, no. 05 (2016): 1650088. http://dx.doi.org/10.1142/s0219498816500882.

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In this paper, we prove that Dedekind–Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a con

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7

Shao, Yong, Sinisa Crvenkovic, and Melanija Mitrovic. "Distributive lattices of Jacobson rings." Publications de l'Institut Math?matique (Belgrade) 100, no. 114 (2016): 87–93. http://dx.doi.org/10.2298/pim1614087s.

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We characterize the distributive lattices of Jacobson rings and prove that if a semiring is a distributive lattice of Jacobson rings, then, up to isomorphism, it is equal to the subdirect product of a distributive lattice and a Jacobson ring. Also, we give a general method to construct distributive lattices of Jacobson rings.

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8

Shao, Yong, Miaomiao Ren, Sinisa Crvenkovic, and Melanija Mitrovic. "The semiring variety generated by any finite number of finite fields and distributive lattices." Publications de l'Institut Math?matique (Belgrade) 98, no. 112 (2015): 45–51. http://dx.doi.org/10.2298/pim150404026s.

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In this paper we study the semiring variety V generated by any finite number of finite fields F1,..., Fk and two-element distributive lattice B2, i.e., V = HSP{B2, F1,..., Fk}. It is proved that V is hereditarily finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1,..., Fk are the only subdirectly irreducible semirings in V.

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9

Katsov, Yefim, Tran Giang Nam, and Jens Zumbrägel. "On simpleness of semirings and complete semirings." Journal of Algebra and Its Applications 13, no. 06 (2014): 1450015. http://dx.doi.org/10.1142/s0219498814500157.

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In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show that the concepts of simpleness, congruence-simpleness, and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive latt

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10

DROSTE, MANFRED, and HEIKO VOGLER. "THE CHOMSKY-SCHÜTZENBERGER THEOREM FOR QUANTITATIVE CONTEXT-FREE LANGUAGES." International Journal of Foundations of Computer Science 25, no. 08 (2014): 955–69. http://dx.doi.org/10.1142/s0129054114400176.

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Weighted automata model quantitative aspects of systems like the consumption of resources during executions. Traditionally, the weights are assumed to form the algebraic structure of a semiring, but recently also other weight computations like average have been considered. Here, we investigate quantitative context-free languages over very general weight structures incorporating all semirings, average computations, lattices. In our main result, we derive the Chomsky-Schützenberger Theorem for such quantitative context-free languages, showing that each arises as the image of the intersection of

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11

Rudeanu, Sergiu, and Dragoş Vaida. "Semirings in Operations Research and Computer Science: More Algebra." Fundamenta Informaticae 61, no. 1 (2004): 61–85. https://doi.org/10.3233/fun-2004-61106.

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We undertake an axiomatic study of certain semirings and related structures that occur in operations research and computer science. We focus on the properties A,I,U,G,Z,L that have been used in the algebraic study of path problems in graphs and prove that the only implications linking the above properties are essentially those already known. On the other hand we extend those implications to the framework of left and right variants of A,I,U,G,Z,L, and we also prove two embedding theorems. Further generalizations refer mainly to semiring-like algebras with a partially defined addition, which is

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12

Shao, Yong, Sinisa Crvenkovic, and Melanija Mitrovic. "The variety of semirings generated by distributive lattices and finite fields." Publications de l'Institut Math?matique (Belgrade) 95, no. 109 (2014): 101–9. http://dx.doi.org/10.2298/pim1409101s.

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A semiring variety is d-semisimple if it is generated by the distributive lattice of order two and a finite number of finite fields. A d-semisimple variety V = HSP{B2, F1,..., Fk} plays the main role in this paper. It will be proved that it is finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1,..., Fk are the only subdirectly irreducible members in it.

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13

Bag, Moumita, Sunil Kumar Maity, and Mridul Kanti Sen. "Completely regular R-semirings and completely regular L-semirings." Quasigroups and Related Systems 32, no. 2(52) (2025): 177–89. https://doi.org/10.56415/qrs.v32.14.

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In [6] it was shown that a semiring is completely regular semiring if and only if it is a b-lattice of completely simple semirings. In this paper, we generalize this concept and introduce completely regular R-semiring, completely regular L-semiring and we show that a semiring is completely regular semiring if and only if it is both a completely regular L-semiring and a completely regular R-semiring. Moreover, we show that a semiring is a completely regular R-semiring (completely regular L-semiring) if and only if it is a b-lattice of completely simple R-semirings (completely simple L-semirings

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14

GUAN, XUECHONG, YONGMING LI, and JUERG KOHLAS. "On conditions for semirings to induce compact information algebras." Mathematical Structures in Computer Science 27, no. 4 (2015): 460–69. http://dx.doi.org/10.1017/s0960129515000225.

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In this paper, we study the relationship between ordering structures on semirings and semiring-induced valuation algebras. We show that a semiring-induced valuation algebra is a complete (resp. continuous) lattice if and only if the semiring is complete (resp. continuous) lattice with respect to the reverse order relation on semirings. Furthermore, a semiring-induced information algebra is compact, if the dual of the semiring is an algebraic lattice.

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15

Borzooei, R. A., M. Shenavaei, A. Di Nola, and O. Zahiri. "On EMV-Semirings." Mathematica Slovaca 69, no. 4 (2019): 739–52. http://dx.doi.org/10.1515/ms-2017-0265.

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Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that a

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16

Biswas, Pronay, Sagarmoy Bag, and Sujit Sardar. "Some new class of ideals in semirings and their applications." Filomat 38, no. 29 (2024): 10223–37. https://doi.org/10.2298/fil2429223b.

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In the present paper, we focus on semirings, which are positive cones of a class of lattice-ordered rings. We establish a lattice isomorphism between semiring l-ideals and ring l-ideals of cancellative l-semirings and its difference l-ring, and from this, we obtain a structure of semiring l-ideals via ring l-ideals in cancellative l-semirings. Smith in [26] defined f-semirings as a suitable class of l-semirings in which one can establish a structure theorem. A new class of f-semirings, namely P-semirings, is defined to focus solely on positive cones of abundant function rings, e.g., C(X). We b

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17

Huang, Kaiqing, Yizhi Chen, and Miaomiao Ren. "Additively orthodox semirings with special transversals." AIMS Mathematics 7, no. 3 (2022): 4153–67. http://dx.doi.org/10.3934/math.2022230.

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<abstract><p>A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding re

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18

Susilowati, Eka, and Ari Suparwanto. "THE BEST SOLUTION FOR INEQUALITIES OF A O CROSS X LOWER THAN X FROM B O DOT X USING HIGH MATRIX RESIDUATION OF IDEMPOTENT SEMIRING." Jurnal Sains Dasar 5, no. 1 (2017): 43. http://dx.doi.org/10.21831/jsd.v5i1.12663.

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Abstract A complete idempotent semiring has a structure which is called a complete lattice. Because of the same structure as the complete lattice then inequality of the complete idempotent semiring can be solved a solution by using residuation theory. One of the inequality which is explained is where matrices A,X,B with entries in the complete idempotent semiring S. Furthermore, introduced dual product , i.e. binary operation endowed in a complete idempotent semirings S and not included in the standard definition of complete idempotent semirings. A solution of inequality can be solved by using

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19

Bhuniya, A. K. "STRUCTURE AND SOME CHARACTERIZATIONS OF LEFT k-CLIFFORD SEMIRINGS." Asian-European Journal of Mathematics 06, no. 04 (2013): 1350056. http://dx.doi.org/10.1142/s1793557113500563.

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Here, being motivated from the works of Zhu, Guo and Shum on left Clifford semigroups, we have introduced left k-Clifford semirings as a generalization of k-Clifford semirings. A k-regular semiring S ∈ 𝕊𝕃+ is a left k-Clifford semiring (left k-semifield) if for all a ∈ S, [Formula: see text] (for all b ∈ S′, [Formula: see text]). Several characteristics for this class of semirings are obtained. Moreover a semiring is left k-Clifford if and only if it is a distributive lattice of left k-semifields.

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20

Sproat, Richard, Mahsa Yarmohammadi, Izhak Shafran, and Brian Roark. "Applications of Lexicographic Semirings to Problems in Speech and Language Processing." Computational Linguistics 40, no. 4 (2014): 733–61. http://dx.doi.org/10.1162/coli_a_00198.

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This paper explores lexicographic semirings and their application to problems in speech and language processing. Specifically, we present two instantiations of binary lexicographic semirings, one involving a pair of tropical weights, and the other a tropical weight paired with a novel string semiring we term the categorial semiring. The first of these is used to yield an exact encoding of backoff models with epsilon transitions. This lexicographic language model semiring allows for off-line optimization of exact models represented as large weighted finite-state transducers in contrast to impli

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21

Di Nola, Antonio, Giacomo Lenzi, and Tran Giang Nam. "Ultramatricial algebras over commutative chain semirings and application to MV-algebras." Forum Mathematicum 32, no. 2 (2020): 287–305. http://dx.doi.org/10.1515/forum-2019-0056.

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AbstractIn this paper, we give a complete description of strongly projective semimodules over a semiring which is a finite direct product of matrix semirings over commutative chain semirings. We then classify ultramatricial algebras over commutative chain semirings by their ordered {\mathrm{SK}_{0}}-groups. Consequently, we get that there is a one-one correspondence between isomorphism classes of ultramatricial algebras A whose {\mathrm{SK}_{0}(A)} is lattice-ordered over a given commutative chain semiring and isomorphism classes of countable MV-algebras.

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22

Tilak, Raj Sharma, and Sharma Ritu. "Prime Fuzzy Ideals of a Gamma Semiring -1." Indian Journal of Science and Technology 16, no. 31 (2023): 2441–46. https://doi.org/10.17485/IJST/v16i31.1333.

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Abstract <strong>Objectives:</strong> The main objective of this article is to introduce some fundamental results of fuzzy ideals and Prime fuzzy ideals of a G􀀀 semiring R . <strong>Methods:</strong> We use some fundamental results of fuzzy ideals and conditions like semi subtractive, centreless etc. for developments of the main results. <strong>Findings:</strong> In this connection, we establish some important results of fuzzy ideals and prime fuzzy ideals of semirings to G􀀀 semirings. <strong>Novelty:</strong> We find a special class of fuzzy ideals of a G􀀀

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23

Sen, M. K., and A. K. Bhuniya. "The Structure of Almost Idempotent Semirings." Algebra Colloquium 17, spec01 (2010): 851–64. http://dx.doi.org/10.1142/s1005386710000799.

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In this paper we introduce the notion of almost idempotent semirings as the semirings with semilattice additive reduct satisfying the identity x + x2 = x2, and characterize eight subclasses of the variety [Formula: see text] of all almost idempotent semirings corresponding to the eight subvarieties of the variety [Formula: see text] of all normal bands. Every almost idempotent semiring S is a distributive lattice of rectangular almost idempotent semirings. Given a semigroup F, the semiring Pf(F) of all finite non-empty subsets of F is almost idempotent precisely when F is a band, and in this c

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24

Gupta, Sugato, and Sujit Kumar Sardar. "Morita invariants of semirings-II." Asian-European Journal of Mathematics 11, no. 01 (2017): 1850014. http://dx.doi.org/10.1142/s1793557118500146.

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The purpose of this paper is to investigate the counterparts of most of our results on Morita invariants of semirings for semimodules. Among others, we obtain the lattice isomorphisms between the set of all ideals and set of all subsemimodules corresponding to a Morita context of semirings. Also, the preservation of congruences between a semiring and a semimodule corresponding to a Morita context of semirings is established.

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25

DOLINKA, IGOR. "IDEMPOTENT DISTRIBUTIVE SEMIRINGS WITH INVOLUTION." International Journal of Algebra and Computation 13, no. 05 (2003): 597–625. http://dx.doi.org/10.1142/s0218196703001614.

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A semiring with involution is a semiring equipped with an involutorial antiasutomorphism as a fundamental operation. The aim of the present paper is to determine the lattice of all varieties of idempotent and distributive semirings with involution. We start with the description of their structure, which is followed by a complete list of all subdirectly irreducibles. We make a heavy use of general results obtained recently by Dolinka and Vinčić [11] on involutorial Płonka sums. Applying these results and some further structural theorems, we construct the considered lattice. It turns out that it

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26

Vechtomov, E. M. "Multiplicatively idempotent semirings with annihilator condition. II." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 6 (June 29, 2025): 21–31. https://doi.org/10.26907/0021-3446-2025-6-21-31.

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The article continues the study of multiplicatively idempotent semirings with the annihilation condition. It is proven that for multiplicatively idempotent semirings with zero the annihilation condition is equivalent to the equalizing property (Theorem 1). New conditions are obtained (Rickart property, properties of a simple spectrum, and others) under which a multiplicatively idempotent semiring is isomorphic to the direct product of a Boolean ring and a generalized Boolean lattice (Theorems 2 and 3). Some other statements have also been proved, examples have been given, and explanatory remar

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27

HAJDINJAK, MELITA, and GAVIN BIERMAN. "Extending relational algebra with similarities." Mathematical Structures in Computer Science 22, no. 4 (2012): 686–718. http://dx.doi.org/10.1017/s0960129511000740.

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In this paper we propose various extensions to the relational model to support similarity-based querying. We build upon the -relation model, where tuples are assigned values from an arbitrary semiring , and its associated positive relational algebra $\text{RA}^{+}_{\mathcal{K}}$. We consider a recently proposed extension to $\text{RA}^{+}_{\mathcal{K}}$ using a monus operation on the semiring to support negative queries, and show how, surprisingly, it fails for important ‘fuzzy’ semirings. Instead, we suggest using a negation operator. We also consider the identities satisfied by the relationa

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28

Mukhopadhyay, P. "Subdirect products of semirings." International Journal of Mathematics and Mathematical Sciences 26, no. 9 (2001): 539–45. http://dx.doi.org/10.1155/s0161171201003696.

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Bandelt and Petrich (1982) proved that an inversive semiringSis a subdirect product of a distributive lattice and a ring if and only ifSsatisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, in its aforesaid representation as a subdirect product, the ring involved can be gradually enriched to a field. Finally, we provide a construction of fullE-inversive semirings, which are subdirect produc

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29

Mondal, Tapas Kumar, and Anjan Kumar Bhuniya. "On the distributive lattices of left k-archimedean semirings." MATHEMATICA 62 (85), no. 2 (2020): 179–88. http://dx.doi.org/10.24193/mathcluj.2020.2.07.

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We introduce the notion of left k-Archimedean semirings which generalize the notion of k-Archimedean semirings, and characterize the semirings which are distributive lattices (chains) of left k-Archimedean semirings.

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Vechtomov, E. M. "About multiplicatively idempotent semirings with annihilator properties." Mathematics and Theoretical Computer Science 2, no. 4 (2025): 24–34. https://doi.org/10.26907/2949-3919.2024.4.24-34.

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We study one class of semirings close to distributive lattices, namely: semirings with commutative idempotent multiplication, satisfying the identity x + 2xy = x. For such semirings, the equalizing properties are investigated (Theorems 7, 8, 10, Proposition 12). The results obtained can be applied to distributive lattices (Propositions 15, 16, 17). The work provides examples and explanatory notes.

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31

Wani, Swapnil, and Kishor Pawar. "On Full k-ideals of a Ternary Semiring." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 1 (2016): 5805–7. http://dx.doi.org/10.24297/jam.v12i1.590.

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In this paper, we generalize the concept of the full -ideals of a semiring to ternary semiring and prove that the set of zeroids annihilator of a right ternary semimodule, and the Jacobson radical of a ternary semiring are all full -ideals of. Also we prove that the set of all full -ideals of a ternary semiring is a complete lattice which is also modular.

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32

Pondělíček, Bedřich. "Inverse semirings and their lattice of congruences." Czechoslovak Mathematical Journal 46, no. 3 (1996): 513–22. http://dx.doi.org/10.21136/cmj.1996.127312.

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Sardar, Sujit Kumar, and Sugato Gupta. "Morita invariants of semirings." Journal of Algebra and Its Applications 15, no. 02 (2015): 1650023. http://dx.doi.org/10.1142/s0219498816500237.

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In this paper we revisit that ideal lattices and congruence lattices are preserved by Morita equivalence of semirings which is originally obtained implicitly by Katsov and his co-authors. This is then used to obtain some Morita invariants for semirings.

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34

Kar, S., S. Purkait, and R. Sarkar. "Birkhoff center of c-semiring." Asian-European Journal of Mathematics 12, no. 01 (2019): 1950003. http://dx.doi.org/10.1142/s1793557119500037.

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In this paper, we introduce the concept of Birkhoff center of a [Formula: see text]-semiring and provide some characterizations of this center. Finally, we prove that the Birkhoff center of a [Formula: see text]-semiring forms a distributive lattice.

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35

Sen, M. K., and M. R. Adhikari. "Onk-ideals of semirings." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 347–50. http://dx.doi.org/10.1155/s0161171292000437.

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Certain types of ring congruences on an additive inverse semiring are characterized with the help of fullk-ideals. It is also shown that the set of all fullk-ideals of an additively inverse semiring in which addition is commutative forms a complete lattice which is also modular.

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Chajda, Ivan, and Helmut Länger. "Basic semirings." Mathematica Slovaca 69, no. 3 (2019): 533–40. http://dx.doi.org/10.1515/ms-2017-0245.

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Abstract Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the referen

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Abuhlail, J. Y., S. N. Il’in, Y. Katsov, and T. G. Nam. "Toward homological characterization of semirings by e-injective semimodules." Journal of Algebra and Its Applications 17, no. 04 (2018): 1850059. http://dx.doi.org/10.1142/s0219498818500597.

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In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.

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Chowdhury, Kanak Ray, Abeda Sultana, Nirmal Kanti Mitra, and A. F. M. Khodadad Khan. "Relation Between Lattice and Semiring." JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES 9, no. 1 (2014): 1339–53. http://dx.doi.org/10.26782/jmcms.2014.07.00006.

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39

Iftikhar, Muhammad, and Tahir Mahmood. "Some results on lattice ordered double framed soft semirings." International Journal of Algebra and Statistics 7, no. 1-2 (2018): 123–40. http://dx.doi.org/10.20454/ijas.2018.1491.

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In this article the concept of lattice ordered double framed soft semirings are introduced and establish some results with examples by using basic operations like union, intersection, "AND" product, "OR" product, restricted union, restricted intersection. Furthermore, lattice ordered double framed soft ideals and lattice ordered idealistic double framed soft semirings are also given with some related examples.

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40

Shao, Yong, and Miaomiao Ren. "On sturdy frame of abstract algebras." Publications de l'Institut Math?matique (Belgrade) 97, no. 111 (2015): 199–210. http://dx.doi.org/10.2298/pim140421001s.

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We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an application, we study the sturdy distributive lattice of lattice ordered groups.

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41

Devi, D. Mrudula, G. Shobha Latha, and T. Padma Praveen. "Properties of Connected Semirings and B-lattice Semirings." International Journal of Mathematics Trends and Technology 52, no. 6 (2017): 370–73. http://dx.doi.org/10.14445/22315373/ijmtt-v52p552.

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42

Jiang, Jing, Lan Shu, and Xinan Tian. "On Generalized Transitive Matrices." Journal of Applied Mathematics 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/164371.

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Transitivity of generalized fuzzy matrices over a special type of semiring is considered. The semiring is called incline algebra which generalizes Boolean algebra, fuzzy algebra, and distributive lattice. This paper studies the transitive incline matrices in detail. The transitive closure of an incline matrix is studied, and the convergence for powers of transitive incline matrices is considered. Some properties of compositions of incline matrices are also given, and a new transitive incline matrix is constructed from given incline matrices. Finally, the issue of the canonical form of a transi

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43

Mondal, Tapas Kumar, and Anjan Kumar Bhuniya. "Semirings which are distributive lattices of $t$-$k$-simple semirings." Tbilisi Mathematical Journal 8, no. 2 (2015): 149–57. http://dx.doi.org/10.1515/tmj-2015-0018.

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44

Valverde-Albacete, Francisco José, and Carmen Peláez-Moreno. "Four-Fold Formal Concept Analysis Based on Complete Idempotent Semifields." Mathematics 9, no. 2 (2021): 173. http://dx.doi.org/10.3390/math9020173.

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Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯-four-fold Formal Concept Analysis (K¯-4FCA) where K¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in duall

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45

Kala, Vítězslav, and Miroslav Korbelář. "Idempotence of finitely generated commutative semifields." Forum Mathematicum 30, no. 6 (2018): 1461–74. http://dx.doi.org/10.1515/forum-2017-0098.

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Abstract We prove that a commutative parasemifield S is additively idempotent, provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

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46

BHARGAVI, YELLA, AKBAR REZAE, TAMMA ESWARLAL та SISTLA RAGAMAYI. "Vague Weak Interior Ideals of Γ-Semirings". Kragujevac Journal of Mathematics 49, № 5 (2024): 711–26. http://dx.doi.org/10.46793/kgjmat2505.711b.

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The notion of a ((complete-) normal) vague weak interior ideal on a (regular) Γ-semiring is deﬁned. It is proved that the set of all vague weak interior ideals forms a complete lattice. Also, a characterization theorem for a regular Γ-semiring in terms of vague weak interior ideals is derived. Another interesting consequence of the main result is that the cardinal of a non-constant maximal element in the set of all (complete-) normal vague weak interior ideals is 2.

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Wang, Aifa, and Yong Shao. "On some varieties of ai-semirings satisfying xp+1 ≈ x." Open Mathematics 16, no. 1 (2018): 913–23. http://dx.doi.org/10.1515/math-2018-0079.

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AbstractThe aim of this paper is to study the lattice of subvarieties of the ai-semiring variety defined by the additional identities$$\begin{array}{} \displaystyle x^{p+1}\approx x\,\,\mbox{and}\,\,zxyz\approx(zxzyz)^{p}zyxz(zxzyz)^{p}, \end{array} $$wherepis a prime. It is shown that this lattice is a distributive lattice of order 179. Also, each member of this lattice is finitely based and finitely generated.

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Mondal, T. "Distributive Lattices of λ-simple Semirings". Iranian Journal of Mathematical Sciences and Informatics 17, № 1 (2022): 47–55. http://dx.doi.org/10.52547/ijmsi.17.1.47.

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Chajda, Ivan, and Helmut Länger. "General coupled semirings of residuated lattices." Fuzzy Sets and Systems 303 (November 2016): 128–35. http://dx.doi.org/10.1016/j.fss.2015.12.009.

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Jipsen, Peter. "From Semirings to Residuated Kleene Lattices." Studia Logica 76, no. 2 (2004): 291–303. http://dx.doi.org/10.1023/b:stud.0000032089.54776.63.

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