Seminar of the Extension of the Mathematical Institute SAS in Košice

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9. 6. 2022, 9:15 Michal Hospodár – Operations on subregular languages and nondeterministic state complexity
 
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12. 5. 2022 Ján Haluška – An ensemble of 6 ordered vector algebras, epimorphic to 6 generalized complex numbers is a generalization of 12-TET
Abstract: A $12$-dimensional linearly ordered vector algebra $\mathscr{W}_{12}$ over $\mathbb{R}$ is inspired with 12-Tone Equal Temperament Tuning system (12-TET) which is generally known in mathematical acoustics. In the paper introduced definition of multiplication $\otimes$ is based on a chain of 85 vectors of the Pythagorean approximation cycle and the notion of octave equivalence. From the operation $\otimes$, it is derived an operation $\oplus$ (operation of transposition, shift of vectors). The operation $\otimes$ is associative, commutative and distributive with respect to $\oplus$. In the paper, there are considered first 6 sub-algebras of $\mathscr{W}_{12}$; $\mathscr{W}_{1} \subset \mathscr{W}_{2} \subset \mathscr{W}_{3} \subset \mathscr{W}_{6} \subset \mathscr{W}_{12}$, $\mathscr{W}_{2} \subset \mathscr{W}_{4} \subset \mathscr{W}_{12}$. We can compute all possible invertible vectors (if they exist) in all ordered sub-algebras of $\mathscr{W}_{12}$. We describe only some interconnections among various substructures of $\mathscr{W}_{12}$, neither its numerous non-mathematical applications. According to results of the paper, we can claim that each tonal European musical composition is algebraically isomorphic with a sequence of lineals (=chords) of vectors (=tones) in $\mathscr{W}_{12}$ in time.
26. 5. 2022 Peter Eliaš – On tensor product
Abstract: We compare construction of tensor product in the category of vector space, where morphisms are linear maps, and the category of complete lattices, where morphisms are join-preserving maps. Tensor product of vector spaces $V$, $W$ is a vector space $T=V\otimes W$, together with a map $\otimes\colon V\times W\to T$, possessing a universal property with respect to all bilinear maps from the cartesian product $V\times W$ into arbitrary vector space $U$. Tensor product of complete lattices $K$, $L$ (in the category where morphisms are join-preserving maps) is the lattice $G$ of all Galois connections between $K$, $L$, together with an appropriate map $\varphi\colon K\times L\to G$. This map possess a universal property with respect to all maps defined of $K\times L$ that preserve joins in each variable separately.