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Anamorphoses and Function Lattices (Multivalued Morphology)
Published in Edward R. Dougherty, Mathematical Morphology in Image Processing, 2018
Let us conclude this section with a brief comment on duality. Starting from the set of the elements of a complete lattice T, reorder them by stating that t1 is smaller than t2 iff t1 > t2 in T. Obviously, this new order generates a new lattice T✶ on the elements of T, where Sup{xi} = Λ xi and inf{xi} = ⋁ xi for any family {xi in T. The lattice T✶ is said to be dual of T and every proposition on T may be associated with a dual version on T✶ by inverting > and <, on the one hand, and Λ and ⋁ on the other hand. Strictly speaking, the mapping T → T✶ is not an anamorphosis, though it preserves the lattice structure, but rather an involution.
Order and Logic
Published in Pascal Hitzler, Anthony Seda, Mathematical Aspects of Logic Programming Semantics, 2016
A non-empty subset A of a partially ordered set (D, ⊑) is called directed if, for all a, b ⊑∈ A, there is c ∈ A with a ⊑ c and b ⊑ c. An element b in an ordered set D is called an upper bound of a subset A of D if we have a ⊑ b for all a ∈ A and is called a least upper bound or supremum of A if b is an upper bound of A satisfying b ⊑ b’ for all upper bounds b’ of A. Of course, by antisymmetry, the supremum, ⊔ A or sup A, of A is unique if it exists. Similarly, one defines lower bound and the greatest lower bound or infimum, ⊓ A or inf A, of a subset A of D. An element x of D is called maximal (minimal) if we do not have x ⊏ y (y ⊏ x) for any element y of D. Given an ordering ⊑ on a set D, we define the dual ordering ⊏d on D by x ⊏d y if and only if y ⊏ x. Lower bounds, greatest lower bounds, etc. in ⊑ correspond to upper bounds, least upper bounds, etc. in ⊏d.
Introduction
Published in John N. Mordeson, Davender S. Malik, Fuzzy Automata and Languages, 2002
John N. Mordeson, Davender S. Malik
The operations ∨ and ∧ are dual in a complete distributive lattice L=(L,∨,∧). Thus the dual automata of lattice automata can be formulated in the following manner.
Modal translation of substructural logics
Published in Journal of Applied Non-Classical Logics, 2020
A formal concept (Ganter & Wille, 1999) is a pair consisting of a stable and a co-stable set such that (and then also ). The formal concept lattice of the frame is the complete lattice of formal concepts, with operations defined by where meets and joins on the right-hand side of the equations are taken in the complete lattice of (co)stable sets, where meets are set intersections and joins are closures of unions. Clearly, the formal concept lattice is isomorphic to the lattice of concept extents (the complete lattice of Galois stable subsets of X) and dually isomorphic to the lattice of concept intents (the complete lattice of the co-stable subsets of Y).
Exploring how writing-to-learn in a mathematics methods course influences preservice teachers’ beliefs
Published in International Journal of Mathematical Education in Science and Technology, 2023
Tye G. Campbell, Tracey Hodges, Sheunghyun Yeo, Erin Rich, Kaleigh Pate
Following the results of the quantitative analysis, we wanted to understand potential reasons why some preservice teachers’ beliefs changed while others’ beliefs did not change. Further, we wanted to explore explanations for any correlations that we found, mitigating the extraneous factors discussed above. We decided to focus our inquiry on preservice teachers’ beliefs about mathematics pedagogy because the reflection prompts dually positioned preservice teachers and learners and teachers of mathematics. This inquiry led us to ask the third research question: How does the reflective nature of preservice teachers’ writing influence how/whether their beliefs about mathematics pedagogy change during an elementary mathematics methods course?
Subordination Tarski algebras
Published in Journal of Applied Non-Classical Logics, 2019
The paper is organised as follows. In Section 2, we basically review some algebraic notions like implicative filters and ideals in Tarski algebras. In Section 3, we will introduce the notions of quasi-modal operator and subordination relation in Tarski algebras. As we will see these two notions are equivalent. In this section, we shall prove that every subordination Tarski algebra is isomorphic to a subordination Tarski algebra of sets. In Section 4, we will introduce two classes of implicative filters in subordination Tarski algebras. We will define the round filters, or Δ-filters. The round filters are a generalisation of the notion of normal or open filters in modal algebras (Sambin & Vaccaro, 1988). We will prove that the family of all round filters of a subordination Tarski algebra is a lattice dually isomorphic to the family of Tarski congruences that preserves, in a certain sense, the relation of subordination. So, we will introduce the class of irreducible round filters. Every maximal round filter, also called end, is an irreducible round filter, but the reciprocal is not valid. The ends are very important in the representation theory developed by de Vries (1962) because they are points of the dual space of a de Vries algebra. In Section 5, we will study the class of topological and monadic subordination Tarski algebras. These classes are a generalisation of the topological and monadic Tarski algebras (Abad, Díaz Varela, & Zander, 2002), and the class of compingent algebras (de Vries, 1962). The principal results of this section is the characterisation of monadic subordination Tarski algebras as topological subordination Tarski algebras where the notions of end and irreducible round filters are equivalent (see Theorem 5.9). As corollary we have that a topological Tarski (Boolean) algebra is a monadic Tarski (Boolean) algebra iff the set of maximal open filters coincide with the set of □-irreducibles filters. We note that all results present in this paper are also valid in subordination (Boolean) algebras.