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..

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

On the complexities of polymorphic stream equation systems, isomorphism of finitary inductive types, and higher homotopies in univalent universes

This thesis is composed of three separate parts. The first part deals with definability and productivity issues of equational systems defining polymorphic stream functions. The main result consists of showing such systems composed of only unary stream functions complete with respect to specifying computable unary polymorphic stream functions. The second part deals with syntactic and semantic notions of isomorphism of finitary inductive types and associated decidability issues. We show isomorphism of so-called guarded types decidable in the set and syntactic model, verifying that the answers coincide. The third part deals with homotopy levels of hierarchical univalent universes in homotopy type theory, showing that the n-th universe of n-types has truncation level strictly n+1

On the complexities of polymorphic stream equation systems, isomorphism of finitary inductive types, and higher homotopies in univalent universes

This thesis is composed of three separate parts. The first part deals with definability and productivity issues of equational systems defining polymorphic stream functions. The main result consists of showing such systems composed of only unary stream functions complete with respect to specifying computable unary polymorphic stream functions. The second part deals with syntactic and semantic notions of isomorphism of finitary inductive types and associated decidability issues. We show isomorphism of so-called guarded types decidable in the set and syntactic model, verifying that the answers coincide. The third part deals with homotopy levels of hierarchical univalent universes in homotopy type theory, showing that the n-th universe of n-types has truncation level strictly n+1

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition

NOTIFICATION!!!

The full content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition