### Lattice Theory (MATH-8109)

A lattice is an abstract structure studied in the mathematical sub-disciplines of order theory & abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) & a unique infimum (also called a greatest lower bound or meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities.
Since the two definitions are equivalent, lattice theory draws on both order theory & universal algebra. Semi lattices include lattices, which in turn include Heyting & Boolean Algebra. These "lattice-like" structures all admit order-theoretic, as well as algebraic descriptions. Lattice theory is based on a single undefined relation, the inclusion relation & considered as younger brother of Group Theory. It is often said that mathematics is a language. If so, group theory provides the proper vocabulary for discussing symmetry. In the same way, lattice theory provides the proper vocabulary for discussing order, & especially systems which are in any sense hierarchies. One might also say that just as group theory deals with permutations, so lattice theory deals with combinations. One difference between the two is that whereas our knowledge of group theory has increased by not more than fifty per cent in the last thirty years, our knowledge of lattice theory has increased by perhaps two hundred per cent in the last ten years. Lattice theory is the study of sets of objects known as lattices. It is an outgrowth of the study of Boolean algebra, & provides a framework for unifying the study of classes or ordered sets in mathematics. The course aims to provide the students with a thorough knowledge of the lattice theory & its applications to mathematics.

Intended Learning Outcomes: Upon successful completion of the course, a PhD scholar will be able to:

1. Derive combinatorial results for various mathematical operations and tasks, such as interaction of graphs and lattices.
2. Apply the techniques to construct new lattices and its applications.
3. Recognize different methods (for example, Boolean function, evaluating boolean expresion)
4. Implement a variety of techniques using appropriate programming (for example, Maple),
5. Effectively use tools to solve problems in their own field of interest.

Prerequisites: Linear Algebra

Contents

1. Elementary Concepts
2. Definition of lattice
3. Some algebraic concepts,
4. Polynomials,
5. Identities,
6. Inequalities
7. Free lattices
8. Special elements
9. Distributive lattices
10. Characterization & representation theorems
11. Polynomials & freeness
12. Congruence relations.
13. Boolean algebra

Recommended Books

1. Blyth, T. S., & Birkhoff, G. (2005). Lattices & ordered algebraic structures. New York: Springer.
2. Davey, B. A., & Priestley, H. A. (2002). Introduction to lattices & order. Cambridge: Cambridge University Press.

Suggested Books

1. Grazer, G. (1971). Lattice theory. New York: W.H. Freeman & Company.
2. Donnellan, T. (1968). Lattice theory. The Netherlands: Elsevier Science Ltd.

RESEARCH PROJECT /PRACTICALS/LABS/ASSIGNMENTS

Research project/assignment based on numerical techniques and algorithms often involve features which require their solutions can be awarded to students during their course work period. Examples of features are

1. Deveolping connectin between grahs and lattices.
2. Finding solution of problems based on boolean algebra
3. Improving existing techniques of sets, lattices

Assessment Criteria

• Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)
• Mid-Term Exam: 30
• Final-Term Exam: 50

Key Dates and Time of Class Meeting

Wednesday                                                     14:00--17:00

Commencement of Classes                                                   April 12, 2023

Mid Term Examination                                                           June 06-10, 2023

Final Term Examination                                                          August 08-12, 2023

Declaration of Result                                                              August , 2023