Matematica | UNIVERSAL ALGEBRA AND CATEGORY THEORY

cod. 0522200054

UNIVERSAL ALGEBRA AND CATEGORY THEORY

	0522200054
	DIPARTIMENTO DI MATEMATICA
	EQF7
	MATHEMATICS
	2022/2023

CURRICULUM GENERAL MATHEMATICS AND DIDACTICS OF MATHEMATICS

	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	SPRING SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/01	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS

EXAMS

	Exam	Date	Session
	ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO	12/06/2023 - 09:00	SESSIONE ORDINARIA
	ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO	03/07/2023 - 09:00	SESSIONE ORDINARIA

	Objectives
	CRITICAL KNOWLEDGE OF THE FUNDAMENTAL LAWS OF ALGEBRA THAT ARE COMMON TO ALL EQUATIONALLY DEFINED CLASSES OF STRUCTURES AND OF THE MAIN TOOLS FOR THEIR GENERAL STUDY. ** EXPECTED LEARNING OUTCOMES: * KNOWLEDGE AND UNDERSTANDING - KNOWLEDGE OF THE CONCEPT OF LATTICE AND ITS MAIN CLASSIFICATIONS. - KNOWLEDGE OF THE GENERAL CONCEPT OF HOMOMORPHISM, SUBALGEBRA, QUOTIENT. - KNOWLEDGE OF THE CONGRUENCE LATTICE OF AN ALGEBRA AND ITS CONNECTIONS WITH THE ALGEBRA ITSELF. - KNOWLEDGE OF THE MAIN CATEGORICAL CONSTRUCTIONS: PRODUCTS, COMPRODUCTS, DIRECT LIMITS AND INVERSE LIMITS. - KNOWLEDGE OF THE MAIN CATEGORICAL CONCEPR: CATEGORY, FUNCTOR, ADJUNCTION, NATURAL TRANSFORMATION. - KNOWLEDGE OF THE CONCEPT OF FREE ALGEBRA AND ALGEBRA OF POLYNOMIALS. * ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING - ABILITY TO RIGOROUSLY REPRODUCE THE PROOFS OF THE MAIN RESULTS OF THE COURSE. - ABILITY TO APPLY THE TECHNIQUES AND TOOLS OF THE COURSE IN SIMILAR CASES. - ABILITY TO CLASSIFY EQUATIONALLY DEFINED CLASSES OF ALGEBRAS ON THE BASIS OF THEIR MAIN PROPERTIES. - ABILITY TO EXTRAPOLATE THE CRUCIAL ASPECTS OF AN ARBITRARY CLASS OF EQUATIONAL STRUCTURES. AUTONOMY OF JUDGMENT THE STUDENT MUST BE ABLE TO CONNECT THE COURSE TOPICS WITH THOSE OF ANALYSIS, ALGEBRA, GEOMETRY AND THEORETICAL COMPUTER THEMES. COMMUNICATION SKILLS THE STUDENT WILL BE ABLE TO HOLD CONVERSATIONS WITH PRECISION AND RIGOR ON ISSUES RELATED TO UNIVERSAL ALGEBRA AND CATEGORY THEORY.

CRITICAL KNOWLEDGE OF THE FUNDAMENTAL LAWS OF ALGEBRA THAT ARE COMMON TO ALL EQUATIONALLY DEFINED CLASSES OF STRUCTURES AND OF THE MAIN TOOLS FOR THEIR GENERAL STUDY.

** EXPECTED LEARNING OUTCOMES:
* KNOWLEDGE AND UNDERSTANDING

- KNOWLEDGE OF THE CONCEPT OF LATTICE AND ITS MAIN CLASSIFICATIONS.
- KNOWLEDGE OF THE GENERAL CONCEPT OF HOMOMORPHISM, SUBALGEBRA, QUOTIENT.
- KNOWLEDGE OF THE CONGRUENCE LATTICE OF AN ALGEBRA AND ITS CONNECTIONS WITH THE ALGEBRA ITSELF.
- KNOWLEDGE OF THE MAIN CATEGORICAL CONSTRUCTIONS: PRODUCTS, COMPRODUCTS, DIRECT LIMITS AND INVERSE LIMITS.
- KNOWLEDGE OF THE MAIN CATEGORICAL CONCEPR: CATEGORY, FUNCTOR, ADJUNCTION, NATURAL TRANSFORMATION.
- KNOWLEDGE OF THE CONCEPT OF FREE ALGEBRA AND ALGEBRA OF POLYNOMIALS.

* ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING

- ABILITY TO RIGOROUSLY REPRODUCE THE PROOFS OF THE MAIN RESULTS OF THE COURSE.
- ABILITY TO APPLY THE TECHNIQUES AND TOOLS OF THE COURSE IN SIMILAR CASES.
- ABILITY TO CLASSIFY EQUATIONALLY DEFINED CLASSES OF ALGEBRAS ON THE BASIS OF THEIR MAIN PROPERTIES.
- ABILITY TO EXTRAPOLATE THE CRUCIAL ASPECTS OF AN ARBITRARY CLASS OF EQUATIONAL STRUCTURES.

** AUTONOMY OF JUDGMENT

THE STUDENT MUST BE ABLE TO CONNECT THE COURSE TOPICS WITH THOSE OF ANALYSIS, ALGEBRA, GEOMETRY AND THEORETICAL COMPUTER THEMES.

** COMMUNICATION SKILLS
THE STUDENT WILL BE ABLE TO HOLD CONVERSATIONS WITH PRECISION AND RIGOR ON ISSUES RELATED TO UNIVERSAL ALGEBRA AND CATEGORY THEORY.

	Prerequisites
	BASIC KNOWLEDGE OF ALGEBRA AND SET THEORY

	Contents
	- CLASSES OF ALGEBRAIC STRUCTURES. (4 HOURS) - SUBALGEBRAS, HOMOMORPHISMS AND CONGRUENCES. (4 HOURS) - LATTICE THEORY (6 HOURS) - CATEGORIES, FUNCTORS (8 HOURS) - PRODUCTS, CO-PRODUCTS, SUB-DIRECT PRODUCTS. (4 HOURS) - LIMITS AND CO-LIMITS (4 HOURS) - NATURAL TRANSFORMATIONS AND ADJUNCTIONS. (6 HOURS) - FREE ALGEBRAS. (4 HOURS) - EQUATIONAL CLASSES (4 HOURS) - HSP THEOREM IN CATEGORICAL LANGUAGE (4 HOURS)

Contents

- CLASSES OF ALGEBRAIC STRUCTURES. (4 HOURS)
- SUBALGEBRAS, HOMOMORPHISMS AND CONGRUENCES. (4 HOURS)
- LATTICE THEORY (6 HOURS)
- CATEGORIES, FUNCTORS (8 HOURS)
- PRODUCTS, CO-PRODUCTS, SUB-DIRECT PRODUCTS. (4 HOURS)
- LIMITS AND CO-LIMITS (4 HOURS)
- NATURAL TRANSFORMATIONS AND ADJUNCTIONS. (6 HOURS)
- FREE ALGEBRAS. (4 HOURS)
- EQUATIONAL CLASSES (4 HOURS)
- HSP THEOREM IN CATEGORICAL LANGUAGE (4 HOURS)

	Teaching Methods
	THE 48-HOUR COURSE TAKE PLACE IN THE II SEMESTER. THE COURSE INCLUDES THEORETICAL LESSONS WITH GROUP DISCUSSIONS (6CFU) IN THE CLASSROOM. DURING THE DISCUSSIONS, STUDENTS (POSSIBLY IN GROUPS) SOLVE THEORETICAL PROBLEMS WHICH WILL THEN BE USED TO ACHIEVE MORE INVOLVED RESULTS. THIS LAST PHASE PROMOTES THE ABILITY TO IMAGINE POSSIBLE STRATEGIES TO FORMALIZE INTUITIONS AND TO BUILD COMPLEX CONCEPTS STARTING FROM BASIC ONES.

Teaching Methods

THE 48-HOUR COURSE TAKE PLACE IN THE II SEMESTER.
THE COURSE INCLUDES THEORETICAL LESSONS WITH GROUP DISCUSSIONS (6CFU) IN THE CLASSROOM. DURING THE DISCUSSIONS, STUDENTS (POSSIBLY IN GROUPS) SOLVE THEORETICAL PROBLEMS WHICH WILL THEN BE USED TO ACHIEVE MORE INVOLVED RESULTS. THIS LAST PHASE PROMOTES THE ABILITY TO IMAGINE POSSIBLE STRATEGIES TO FORMALIZE INTUITIONS AND TO BUILD COMPLEX CONCEPTS STARTING FROM BASIC ONES.

	Verification of learning
	THE EXAM AIMS TO EVALUATE THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS INTRODUCED DURING THE LECTURES, AS WELL AS THE ACCURACY AND INDEPENDENCE IN USING SUCH TOOLS. THE EXAMINATION CONSISTS OF AN ORAL INTERVIEW (ABOUT 45 MINUTES) WHERE IT WILL BE EVALUATED THE KNOWLEDGE ACQUIRED ON BASIC AND MOST ADVANCED CONCEPTS IN UNIVERSAL ALGEBRA. STUDENTS MUST FIRST DEMONSTRATE THAT THEY KNOW THE CONCEPTS (DEFINITIONS) COVERED DURING THE COURSE AND THAT THEY UNDERSTOOD THEM, SHOWING THAT THEY CAN INDEPENDENTLY BUILD EXAMPLES. LATER THE QUESTIONS WILL BE AIMED AT UNDERSTANDING IF STUDENTS KNOW HOW TO USE THOSE CONCEPTS AND DEFINITIONS AND KNOW THE FUNDAMENTAL PROPERTIES SEEN DURING THE COURSE (THEOREMS). ONLY IF BOTH THE PREVIOUS PARTS ARE SUCCESSFULLY OVERCOME THE REASONS WHY THESE PROPERTIES HOLDS WILL BE DISCUSSED (DEMONSTRATIONS). THE FINAL MARK IS UP TO THIRTY. LAUDE WILL BE ATTRIBUTED TO STUDENTS THAT PROVE THEMSELVES TO BE ABLE TO INDEPENDENTLY USE THE KNOWLEDGE AND SKILLS ACQUIRED ON THE MOST ADVANCED TOPICS OF THE COURSE, AND ARE ABLE TO FIND CONNECTIONS WITH CONTEXTS DIFFERENT THAN THOSE PRESENTED IN THE LECTURES.

Verification of learning

THE EXAM AIMS TO EVALUATE THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS INTRODUCED DURING THE LECTURES, AS WELL AS THE ACCURACY AND INDEPENDENCE IN USING SUCH TOOLS.

THE EXAMINATION CONSISTS OF AN ORAL INTERVIEW (ABOUT 45 MINUTES) WHERE IT WILL BE EVALUATED THE KNOWLEDGE ACQUIRED ON BASIC AND MOST ADVANCED CONCEPTS IN UNIVERSAL ALGEBRA. STUDENTS MUST FIRST DEMONSTRATE THAT THEY KNOW THE CONCEPTS (DEFINITIONS) COVERED DURING THE COURSE AND THAT THEY UNDERSTOOD THEM, SHOWING THAT THEY CAN INDEPENDENTLY BUILD EXAMPLES. LATER THE QUESTIONS WILL BE AIMED AT UNDERSTANDING IF STUDENTS KNOW HOW TO USE THOSE CONCEPTS AND DEFINITIONS AND KNOW THE FUNDAMENTAL PROPERTIES SEEN DURING THE COURSE (THEOREMS). ONLY IF BOTH THE PREVIOUS PARTS ARE SUCCESSFULLY OVERCOME THE REASONS WHY THESE PROPERTIES HOLDS WILL BE DISCUSSED (DEMONSTRATIONS).

THE FINAL MARK IS UP TO THIRTY. LAUDE WILL BE ATTRIBUTED TO STUDENTS THAT PROVE THEMSELVES TO BE ABLE TO INDEPENDENTLY USE THE KNOWLEDGE AND SKILLS ACQUIRED ON THE MOST ADVANCED TOPICS OF THE COURSE, AND ARE ABLE TO FIND CONNECTIONS WITH CONTEXTS DIFFERENT THAN THOSE PRESENTED IN THE LECTURES.

	Texts
	GEORGE M. BERGMAN. AN INVITATION TO GENERAL ALGEBRA AND UNIVERSAL CONSTRUCTIONS. SPRINGER 2015 S. BURRIS, H. P. SANKAPPANAVAR - A COURSE ON UNIVERSAL ALGEBRA. ONLINE HTTP://WWW.MATH.HAWAII.EDU/~RALPH/CLASSES/619/UNIV-ALGEBRA.PDF G. GRÄTZER. UNIVERSAL ALGEBRA. SECOND EDITION. SPRINGER 2008.