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Minion Cores of Clones
Kapytka, Maryia ; Barto, Libor (advisor) ; Zhuk, Dmitrii (referee)
This thesis provides a classification of the minion homomorphism preordering and minion cores within a class of multi-sorted Boolean clones. These clones can be described as those clones defined on the set {0, 1}k = {0, 1} × {0, 1} × · · · × {0, 1}, where the clone operations act component-wise on the k-tuples, which are determined by multi-sorted unary or binary relations. The second chapter of this thesis focuses on presenting the key findings. We introduce specific minion cores and establish the preordering among them. Furthermore, we prove that each clone falling under the aforementioned type is equivalent to one of these minion cores.
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Generalizing CSP-related results to infinite algebras
Olšák, Miroslav ; Barto, Libor (advisor) ; Zhuk, Dmitrii (referee) ; Pinsker, Michael (referee)
The recent research on constraint satisfaction problems (CSPs) on fixed finite templates provided useful tools for computational complexity and universal algebra. However, the research mainly focused on finite relational structures, and consequently, finite algebras. We pursue a generalization of these tools and results into the domain of infinite algebras. In particular, we show that despite the fact that the Maltsev condition s(r, a, r, e) = s(a, r, e, a) does not characterize Taylor algebras (i.e., algebras that satisfy a nontrivial idem- potent Maltsev condition) in general, as it does in the finite case, there is another strong Maltsev condition characterizing Taylor algebras, and s(r, a, r, e) = s(a, r, e, a) characterizes another interesting broad class of algebras. We also provide a (weak) Maltsev condition for SD(∧) algebras (i.e., algebras that satisfy an idem- potent Maltsev condition not satisfiable in a module). Beyond Maltsev conditions, we study loop lemmata and, in particular, reprove a well known finite loop lemma by two different general (infinite) approaches.
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