The book “Logical Algebras,” authored by Dr. Rajab Ali Borzooei, a professor at Shahid Beheshti University (SBU)'s Faculty of Mathematical Sciences, along with Mona Aaly Kologani and Seyed Hamid Reza Varasteh Kashani, has been released recently by SBU Press in 222 pages.
Logical algebraic structures are crucial in various fields of mathematics and computer science. For example, algebraic structures like Boolean algebra form the foundation of classical logic or binary logic. These structures aid in a better understanding of propositional logic and first-order logic, allowing logical principles to be formally expressed. Boolean algebra is widely used in the design and analysis of logical circuits and digital systems. By describing any logical circuit with Boolean algebraic equations, engineers can design and analyze complex digital systems. Logic and Boolean algebra play a vital role in designing programming languages and analyzing programs, as program analysis tools and compilers use these structures for code optimization and verification. Additionally, these structures are employed in the development of expert systems and artificial intelligence for automated reasoning and processing, and in cryptography and information security for designing encryption algorithms and analyzing system security. They help in creating more secure and efficient algorithms. Essentially, logical algebraic structures provide powerful tools for analyzing and designing complex systems in mathematics, computer science, and engineering, allowing us to solve logical and computational problems accurately and formally.
In this book, some algebraic structures such as residuated lattices, MTL-algebras, BL-algebras, and MV-algebras are introduced across five chapters. All of these are generalizations of Boolean algebra and represent the algebraic expressions of various non-classical logics, like Łukasiewicz logic and fuzzy logic. These logical algebras and the relationships between them are thoroughly examined in this book. Each chapter includes numerous examples to help the reader grasp these algebraic structures. Additionally, after introducing and stating the main features of the algebra, these features are carefully analyzed. The most important subalgebras of these structures, known as filters and ideals, are also defined, their relationships explored, and diagrams provided for better understanding. The book is intended for students of mathematics, computer science, artificial intelligence, engineers, and enthusiasts.
Students and interested individuals can purchase this book from the sales office of the SBU Press located on the ground floor of the Information Technology (IT) building, or order it online via the university’s website. The electronic version of the book is also available for purchase on platforms such as "Fidibo”, “Booket” and “Taaghche".