On zero divisors, invertibility and rank of matrices over commutative semirings

Poluprsten sa nulom i jedinicom je algebarska struktura, koja generališe prsten. Naime, dok prsten u odnosu na sabiranje čini grupu, poluprsten čini samo monoid. Nedostatak oduzimanja čini ovu strukturu znatno težom za istraživanje od prstena. Predmet izučavanja u ovoj tezi predstavljaju matrice nad komutativnim poluprstenima (sa nulom i jedinicom). Motivacija za istraživanje je sadržana u pokušaju da se ispita koje se osobine za matrice nad komutativnim prstenima mogu proširiti na matrice nad komutativnim poluprstenima, a takodje, što je tesno povezano sa ovim pitanjem, kako se svojstva modula nad prstenima prenose na polumodule nad poluprstenima. Izdvajaju se tri tipa dobijenih rezultata. Najpre se proširuju poznati rezultati, koji se tiču dimenzije prostora n-torki elemenata iz nekog poluprstena na drugu klasu poluprstena od do sada poznatih i ispravljaju neke greške u radu drugih autora. Ovo je pitanje u tesnoj vezi sa pitanjem invertibilnosti matrica nad poluprstenima. Drugi tip rezultata se tiče ispitivanja delitelja nule u poluprstenu svih matrica nad komutativnim poluprstenima i to posebno za klasu inverznih poluprstena (to su poluprsteni u kojima postoji neka vrste uopshtenog inverza u odnosu na sabiranje). Zbog nepostojanja oduzimanja, ne može se koristiti determinanta, kao što je to u slučaju matrica nad komutativnim prstenima, ali, zbog činjenice da su u pitanju inverzni poluprsteni, moguće je definisati neku vrstu determinante u ovom slučaju, što omogućava formulaciju odgovorajućih rezultata u ovom slučaju. Zanimljivo je da se za klase matrica za koje se dobijaju rezultati, levi i desni delitelji nule mogu razlikovati, što nije slučaj za komutativne prstene. Treći tip rezultata tiče se pitanja uvodjenja novog ranga za matrice nad komutativnim poluprstenima...Semiring with zero and identity is an algebraic structure which generalizes a ring. Namely, while a ring under addition is a group, a semiring is only a monoid. The lack of substraction makes this structure far more difficult for investigation than a ring. The subject of investigation in this thesis are matrices over commutative semirings (wiht zero and identity). Motivation for this study is contained in an attempt to determine which properties for matrices over commutative rings may be extended to matrices over commutative semirings, and, also, which is closely connected to this question, how can the properties of modules over rings be extended to semimodules over semirings. One may distinguish three types of the obtained results. First, the known results concerning dimension of spaces of n-tuples of elements from a semiring are extended to a new class of semirings from the known ones until now, and some errors from a paper by other authors are corrected. This question is closely related to the question of invertibility of matrices over semirings. Second type of results concerns investigation of zero divisors in a semiring of all matrices over commutative semirings, in particular for a class of inverse semirings (which are those semirings for which there exists some sort of a generalized inverse with respect to addition). Because of the lack of substraction, one cannot use the determinant, as in the case of matrices over commutative semirings, but, because of the fact that the semirings in question are inverse semirings, it is possible to define some sort of determinant in this case, which allows the formulation of corresponding results in this case. It is interesting that for a class of matrices for which the results are obtained, left and right zero divisors may differ, which is not the case for commutative rings. The third type of results is about the question of introducing a new rank for matrices over commutative semirings..

The unitary Cayley graph of a semiring

We study the unitary Cayley graph of a matrix semiring. We find bounds for its diameter, clique number and independence number, and determine its girth. We also find the relationship between the diameter and the clique number of a unitary Cayley graph of a semiring $S$ and a matrix semiring over $S$

Special Quasi Dual Numbers and Groupoids

In this book the authors introduce a new notion called special quasi dual number, x = a + bg. Among the reals – 1 behaves in this way, for (– 1)2 = 1 = – (– 1). Likewise –I behaves in such a way (– I)2 = – (– I). These special quasi dual numbers can be generated from matrices with entries from 1 or I using only the natural product ×n. Another rich source of these special quasi dual numbers or quasi special dual numbers is Zn, n a composite number. For instance 8 in Z12 is such that 82 = 64 = – 8(mod 12) = 4(mod 12). In chapter two we introduce the notion of special quasi dual numbers. The notion of higher dimensional special quasi dual numbers are introduced in chapter three of this book. We using the dual numbers and special dual like numbers with special quasi dual numbers construct three types of mixed special quasi numbers and discuss their properties

Algebraic Structures using Natural Class of Intervals

This book has eleven chapters. Chapter one describes all types of natural class of intervals and the arithmetic operations on them. Chapter two introduces the semigroup of natural class of intervals using R or Zn and study the properties associated with them. Chapter three studies the notion of rings constructed using the natural class of intervals. Matrix theory using the special class of intervals is analyzed in chapter four of this book. Chapter five deals with polynomials using interval coefficients. New types of rings of natural intervals are introduced and studied in chapter six. The notion of vector space using natural class of intervals is built in chapter seven. In chapter eight fuzzy natural class of intervals are introduced and algebraic structures on them is built and described. Algebraic structures using natural class of neutrosophic intervals are developed in chapter nine.Chapter ten suggests some possible applications. The final chapter proposes over 200 problems of which some are at research level and some difficult and others are simple.Comment: 170 pages; Published by The Educational Publisher Inc in 201

สมบัติเชิงพีชคณิตของผลคูณโครเนคเคอร์แบบบล็อกและตัวดำเนินการเวกเตอร์แบบบล็อกสำหรับเมทริกซ์เหนือกึ่งริงสลับที่

บทคัดย่อ เราขยายแนวคิดของผลคูณโครเนคเคอร์ไปสู่ผลคูณโครเนคเคอร์แบบบล็อกสำหรับเมทริกซ์เหนือกึ่งริงสลับที่  เราได้ว่าผลคูณดังกล่าวเข้ากันได้กับการบวกเมทริกซ์ การคูณเมทริกซ์ด้วยสเกลาร์ การคูณเมทริกซ์แบบปรกติ การสลับเปลี่ยน และรอยเมทริกซ์  สมบัติเชิงพีชคณิตหลายอย่างของเมทริกซ์ เช่น ความสมมาตร การหาผกผันได้ ภาวะคล้าย สมภาค การทำเป็นเมทริกซ์ทแยงมุมได้ ถูกรักษาไว้ภายใต้ผลคูณโครเนคเคอร์แบบบล็อก นอกจากนี้เราพิจารณาความสัมพันธ์ระหว่างผลคูณดังกล่าวกับตัวดำเนินการเวกเตอร์แบบบล็อก ความสัมพันธ์ดังกล่าวสามารถนำไปลดรูปสมการเมทริกซ์เชิงเส้นให้อยู่ในรูปสมการเวกเตอร์-เมทริกซ์อย่างง่าย  - - -  Algebraic Properties of the Block Kronecker Product and a Block Vector-Operator for Matrices over a Commutative Semiring  ABSTRACT We extend the notion of Kronecker product to the block Kronecker product for matrices over a commutative semiring. It turns out that this matrix product is compatible with the matrix addition, the scalar multiplication, the usual multiplication, the transposition, and the traces. Certain algebraic properties of matrices, such as symmetry, invertibility, similarity, congruence, diagonalizability, are preserved under the block Kronecker product. In addition, we investigate a relation between this matrix product and a block vector-operator. Such relation can be applied to reduce certain linear matrix equations to simple vector-matrix equations

Linear Algebra and Smarandache Linear Algebra

The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite prime fields, which is not properly represented or analyzed in linear algebra books

Special Dual like Numbers and Lattices

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b are positive reals greater than or equal to one we see powers of x diverge on; and every power of x is also a special dual like number, with very large a and b. On the other hand if a and b are positive reals lying in the open interval (0, 1) then we see the higher powers of x may converge to 0. Another rich source of idempotents is the Neutrosophic number I, as I2 = I. We build several types of finite or infinite rings using these Neutrosophic numbers

Natural Product Xn on matrices

This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research level.Comment: 342; published by Zip publishing, 201