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Journal of Algebra and its Applications, ISSN 0219-4988, 01/2019, Volume 18, Issue 1
We explore algebraic properties of noncommutative frames. The concept of noncommutative frame is due to Le Bruyn, who introduced it in connection with... 
Frames | Heyting algebra | noncommutative structure | skew lattice | MATHEMATICS | MATHEMATICS, APPLIED | Mathematics - Rings and Algebras
Journal Article
Journal of Algebra, ISSN 0021-8693, 01/2020, Volume 542, pp. 65 - 92
In this paper we discuss and characterize several set-theoretic solutions of the Yang-Baxter equation obtained using skew lattices, an algebraic structure that... 
Yang-Baxter equation | Noncommutative lattice | Set-theoretic solutions | Skew lattice | Distributive | MATHEMATICS | SEMI-BRACES
Journal Article
Semigroup Forum, ISSN 0037-1912, 10/2019, Volume 99, Issue 2, pp. 522 - 523
In this corrigendum, we correct the statement and the proof of Proposition 4.11, where the word “lower” is stated instead of “upper,” and vice versa. 
Mathematics | Algebra
Journal Article
Ars Mathematica Contemporanea, ISSN 1855-3966, 2017, Volume 12, Issue 1, pp. 37 - 50
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... 
Non-commutative algebra | Intuitionistic logic | Heyting algebras | Skew lattices | MATHEMATICS | MATHEMATICS, APPLIED | non-commutative algebra | intuitionistic logic | LATTICES | BOOLEAN-ALGEBRAS | STONE DUALITY
Journal Article
Journal of Algebra and its Applications, ISSN 0219-4988, 06/2018
J. Algebra Appl. 18, no. 6 (2019) 1950107 In 1702.04949 noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a... 
Heyting algebra | skew lattice | Noncommutative topos | noncommutative frame | Lawvere–Tierney topology
Journal Article
Algebra universalis, ISSN 0002-5240, 6/2015, Volume 73, Issue 3, pp. 369 - 390
We establish a connection between skew Boolean algebras and Church algebras. We prove that the set of all semicentral elements in a right Church algebra forms... 
Algebra | Church algebra | decomposition operator | discriminator variety | factor congruence | Mathematics | Primary: 03G10 | skew Boolean algebra | Secondary: 08B26 | MATHEMATICS | RINGS | LATTICES | STONE DUALITY
Journal Article
Semigroup Forum, ISSN 0037-1912, 4/2016, Volume 92, Issue 2, pp. 361 - 376
Skew lattices are non-commutative generalizations of lattices, and the cosets are the building blocks of skew lattices. Every skew lattice embeds into a direct... 
Regular Band | Algebra | Block Form | Mathematics | Decomposition Theorem | Left Coset | Triangular Matrice | MATHEMATICS | CLASSIFICATION | MATRICES | RINGS | BANDS
Journal Article
Communications in Algebra, ISSN 0092-7872, 09/2012, Volume 40, Issue 9, pp. 3288 - 3307
Let R be a ring whose set of idempotents E(R) is closed under multiplication. When R has an identity 1, E(R) is known to lie in the center of R, thus forming a... 
Skew Boolean algebra | Class | Secondary 06F05 | Idempotent element | Normal band | Primary 16U80 | D-Class | MATHEMATICS | SKEW LATTICES
Journal Article
Houston Journal of Mathematics, ISSN 0362-1588, 2013, Volume 39, Issue 1, pp. 73 - 109
We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a... 
Stone spaces | Duality | Skew Boolean algebras | MATHEMATICS | LATTICES | skew Boolean algebras | duality | RINGS
Journal Article
Algebra universalis, ISSN 0002-5240, 10/2011, Volume 66, Issue 1, pp. 99 - 113
Skew lattices are a non-commutative generalization of lattices. In the past 20 years, several varieties of skew lattices have been introduced. In the present... 
Primary: 06A11 | Secondary: 06F05 | skew lattice | Algebra | cancellative | strongly symmetric | variety | Mathematics | symmetric | MATHEMATICS | RINGS | BOOLEAN-ALGEBRAS | Universities and colleges
Journal Article
Journal of Applied Logic, ISSN 1570-8683, 09/2013, Volume 11, Issue 3, pp. 253 - 265
A recent study of the override and update operations defined on sets of partial functions placed both operations within the algebraic context of a certain... 
Skew Boolean algebra | Update | Override | Skew lattice | Override Update | MATHEMATICS, APPLIED | SUBTRACTIVE VARIETIES | RINGS | COMPUTER SCIENCE, THEORY & METHODS | BOOLEAN-ALGEBRAS | LOGIC | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | Algebra | Boolean algebra | Equivalence | Logic | Mathematical analysis | Lattices
Journal Article
Linear Algebra and Its Applications, ISSN 0024-3795, 2007, Volume 426, Issue 1, pp. 204 - 213
We prove that any pure regular band of matrices admits a simultaneous LU decomposition in the standard form. In case that such a band forms a double band... 
Idempotent | Semigroup | Band | Matrix | Skew lattice | LU decomposition | idempotent | MATHEMATICS, APPLIED | skew lattice | SKEW LATTICES | RINGS | semigroup | band | matrix | BANDS
Journal Article
Communications in Algebra, ISSN 0092-7872, 12/2006, Volume 35, Issue 1, pp. 243 - 247
In the present note we study internal decompositions of skew lattices. This provides further insight into the structure of skew lattices. Special attention is... 
20M17 | Band | Semigroup | 06A06 | 20M25 | Skew lattice | MATHEMATICS | semigroup | skew lattice | band | RINGS
Journal Article
Semigroup Forum, ISSN 0037-1912, 12/2011, Volume 83, Issue 3, pp. 395 - 411
Skew lattices are a noncommutative generalization of lattices. In the paper we study the varieties of symmetric, strongly symmetric and cancellative skew... 
Coset | Algebra | Noncommutative lattice | Cancellation | Mathematics | Index | Skew lattice | Symmetry | Strong symmetry | MATHEMATICS | ALGEBRAS | Laws, regulations and rules | Universities and colleges
Journal Article
Topology and its Applications, ISSN 0166-8641, 08/2013, Volume 160, Issue 12, pp. 1423 - 1438
We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves... 
Stone duality | Non-commutative algebra | Sheaves over a spectral space | Priestley duality | Sheaves over a Priestley space | Skew lattice | MATHEMATICS | MATHEMATICS, APPLIED | RINGS | REPRESENTATION | SKEW BOOLEAN-ALGEBRAS | DISTRIBUTIVE LATTICES
Journal Article
Semigroup Forum, ISSN 0037-1912, 10/2006, Volume 73, Issue 2, pp. 267 - 272
Skew lattices form a class of non-commutative lattices. Spinks' Theorem [Matthew Spinks, On middle distributivity for skew lattices, Semigroup Forum 61 (2000),... 
Mathematics | Algebra | MATHEMATICS | SKEW LATTICES
Journal Article
Order, ISSN 0167-8094, 3/2011, Volume 28, Issue 1, pp. 9 - 32
Distributive lattices are well known to be precisely those lattices that possess cancellation: $x \lor y = x \lor z$ and $x \land y = x \land z$ imply y = z.... 
Geometry | Distributivity | Convex and Discrete Geometry | 06A11 | Cancellation | Mathematics | Theory of Computation | 06F05 | Variety | Skew lattice | 03G10 | MATHEMATICS | RINGS | BOOLEAN-ALGEBRAS
Journal Article
International Journal of Algebra and Computation, ISSN 0218-1967, 11/2011, Volume 21, Issue 7, pp. 1097 - 1110
Rings of matrices whose idempotents are closed under multiplication are studied and those rings that are maximal subject to certain constraints are... 
idempotent element | normal band | skew lattice | Matrix ring
Journal Article
Algebra universalis, ISSN 0002-5240, 9/2005, Volume 53, Issue 4, pp. 471 - 479
In [6] J. Leech introduced skew lattices in rings. In the present paper we study skew lattices in rings of matrices. We prove that every symmetric, normal skew... 
20M17 | skew lattice | Algebra | 20M25 | semigroup | Mathematics | band | 06A06 | matrix | 16S36 | Band | Semigroup | Matrix | Skew lattice | MATHEMATICS | BANDS
Journal Article
International Journal of Algebra and Computation, ISSN 0218-1967, 11/2011, Volume 21, Issue 7, p. 1097
To access, purchase, authenticate, or subscribe to the full-text of this article, please visit this link: http://dx.doi.org/10.1142/S0218196711006856 
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