MAM Seminars Autumn term 2017

Higher seminars in the subject Mathematics/Applied
Mathematics, Autumn
term 2017.

School of Education, Culture and Communication (UKK), Mälardalen
University.

**October 25, 2017, Wednesday, 15.30-16.30**

**Speaker: Melanija Mitrović,
University of Niš, Serbia**

__Title:__** **Semilattices
of Archimedean Semigroups

The Stockholm Logic Seminar

Organisers: Erik Palmgren (chair), Peter LeFanu
Lumsdaine, Per
Martin-Löf (emeritus)

November 15, 10.00–11.45

**Speaker: Melanija Mitrović,
University of Niš, Serbia**

__Title:__ An introduction to a development of the theory of constructive semigroups with apartness

http://logic.math.su.se/seminar/

**ALGEBRA AND GEOMETRY SEMINAR**

**Department of Mathematics, Uppsala University**

Usually: Tuesday's 15.15-17.00 in 64119

** **

**December
12, 2017, Tuesday, 15.15-17.00**

**Speaker:
Melanija Mitrović, University of Niš, Serbia**

__Title: __**Constructive semigroups
with apartness - a new approach to semigroup
theory**

The
theory of constructive semigroups with apartness are **a new
approach** to semigroup
theory, and **not **a new class of semigroups.
Of course, our work is partly
inspired by classical semigroup theory, but, on the other hand, it is
distinguished from it by two significiant aspects: first, we use intuitionistic logic rather than classical,
secondly, our work is based on the notion of apartness (between
elements,
elements and sets). In short, framework of our work is constructive
mathematics
- roughly, mathematics with
intuitionistics logic. Constructive mathematics is not unique notion.
Principle
trends include the following varieties: INT, RUSS, BISH.
Constructive mathematics in our work (and in
this talk) is Errett Bishop - style constructive mathematics, **BISH**,
[2].** **Despite the fact that
constructive algebra
is (relatively) old
discipline (developed among others by L. Kronecker, van der Waerden, A.
Heyting), it is, compared with
constructive analysis and topology, still of modest size.
Following [1],
the principal novelty in treating basic
algebraic structures constructively is
that apartness becomes a fundamental
notion, i.e. one
axiomatizes rings, groups, and fields with apartness**. **The main aim of our work
within constructive semigroups with apartness (was,
is, and) will be to
give a
little progress in that direction. Although the lecture will be based on material given in [4,5], it is, by
no means an attempt to give a complete overview of our existing
results.
Important sourse of ideas and notions of
our work is [3]. An example of application(s) of these ideas can be
found in
[6]. The standard reference for constructive algebra is [7].

**References:**

1.
M.
J. Beeson: *Foundations of
Constructive Mathematics*, Springer-Verlag, 1985.

2.
E.
Bishop: *Foundations of Constructive
Analysis*, McGraw-Hill, New York, 1967.

3.
D.
S. Bridges, L. S. Vita: *Apartness
and Uniformity - A Constructive
Development*, CiE series on Theory and Applications of
Computability,
Springer, 2011.

4.
S.
Crvenković, M. Mitrović,
D. A. Romano: Semigroups
with Apartness, Mathematical Logic
Quarterly, 59 (6), 2013, 407-414.

5.
S.
Crvenković, M. Mitrović,
D. A. Romano: Basic
Notions of (Constructive) Semigroups
with Apartness,
Semigroup Forum, June 2016, Volume 92,
Issue 3, 659-674.

6.
H.
Geuvers, R. Pollack, F. Wiedijk, J. Zwanenburg: A
Constructive Algebraic Hierarchy in Coq, J.
Symbolic Computation (2002) 34, 271-286.

7.
R.
Mines, F. Richman, W. Ruitenburg: *A Course of Constructive Algebra*,
Springer-Verlag, New York, 1988.

**http://www2.math.uu.se/~mazor/seminar.html**