# Computational Algebra

## Objectives

At the end of this course, the student will have acquired knowledge, skills and competencies in symbolic computation and automated theorem proving. It is intended that students be able to: - Understand the main algorithms of computational algebra; - Know some algorithms of automated theorem provers; - Know some tools for experimental mathematics and its use in modeling and discovery of mathematical results.

## General characterization

##### Code

12914

##### Credits

9.0

##### Responsible teacher

António José Mesquita da Cunha Machado Malheiro

##### Hours

Weekly - 4

Total - 42

##### Teaching language

Inglês

### Prerequisites

Knowledge of algebra.

### Bibliography

- J. M. Howie, Fundamentals of semigroup theory, London Mathematical Society, 1996.
- J. Dixon and B. Mortimer, Permutation groups, Springer, 1996.
- O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009.
- GAP tutorial. https://www.gap-system.org/Manuals/doc/tut/chap0.html
- http://proverx.com/login.php

### Teaching method

Classes are theoretical/practical with oral presentation of concepts, methodologies, and examples,

complemented with problem-solving. Specific student difficulties will be addressed during classes or in individual sessions scheduled with the professor. Continuous assessment is based on two tests. If a student does not obtain approval through continuous assessment he can try it in an additional assessment.

### Evaluation method

1. CONTINUOUS EVALUATION

The continuous evaluation consists of carrying out, during the academic period, two in-person and/or distance tests on the Moodle platform, each one quoted from 0 to 20 values (rounded to one decimal place).

All students who, at the time of the test, are enrolled in the Curricular Unit can take any test.

2. APPROVAL AND FINAL CLASSIFICATION

Let T1 and T2 be the classifications obtained in the 1st and 2nd tests, respectively, rounded to decimals.

The student''s final grade, CF, is obtained by rounding up to the units of (0.5 × T1 + 0.5 × T2). Approval to the UC occurs whenever CF ≥ 10. If CF is below 10 the student fails.

3. EXAM

All students enrolled in the UC can take the exam - see point 1.

In this case, the exam grade (NE) replaces the test''s grade to obtain the final grade, CF. The final grade, CF, of the student, is obtained by rounding to the units of NE. Approval to the UC occurs whenever CF ≥ 10. If CF is below 10 the student fails.

4. GRADE IMPROVEMENT

All students wishing to present themselves with a grade improvement must comply with the legal registration formalities for this purpose (information at the Academic Services). The classification of the improvement exam is obtained as indicated in 4. If this result is higher than the one previously obtained in the course, it will be taken as the final grade. Otherwise, there is no improvement in the final grade.

## Subject matter

1. Generalities about semigroups, Green''s relations and the D-class structure, and Rees''s theorem.

Transformation semigroups and permutation groups.

2. Treatment of permutation groups, transformation semigroups, matrices and vector spaces by

programming in symbolic computation.

3. Birkhoff''s theorem for semigroup varieties.

4. Automated theorems provers: The given clause algorithm, ordering of terms, exploration of conjectures (proofs and counterexamples) in varieties of semigroups, groups, rings and quasi-groups. Proof sketches.

5. Explore recent algebra articles trying to prove new theorems (reciprocal, generalizations, analogs in other varieties, etc.).

## Programs

Programs where the course is taught: