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Free Heyting algebras: revisited
Free Heyting algebras: revisited

... goal of generalizing a method of constructing free algebras for varieties axiomatized by rank 1 axioms to the case of rank 0-1 axioms, we consider the case of Heyting algebras (intuitionistic logic, which is of rank 0-1 for f =→). In particular, we construct free Heyting algebras. For an extension ...
Automatic Continuity from a personal perspective Krzysztof Jarosz www.siue.edu/~kjarosz
Automatic Continuity from a personal perspective Krzysztof Jarosz www.siue.edu/~kjarosz

Connectedness and local connectedness of topological groups and
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... connected and locally connected topological groups. It is easy to verify (see Section 6 of [6]) that both F Γ(X) and AΓ(X), respectively the free topological and the free Abelian topological groups on a space X in the sense of Graev are connected iff X is connected. The situation in the case of the ...
ON THE OPPOSITE OF THE CATEGORY OF RINGS
ON THE OPPOSITE OF THE CATEGORY OF RINGS

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... Since 1 is added to –2n, subtract 1 from both sides to undo the addition. Since n is multiplied by –2, divide both sides by –2 to undo the multiplication. Change ≥ to ≤. ...
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The First Isomorphism Theorem
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... I need to check that this map is well-defined. The point is that a given coset gH may in general be written as g ′ H, where g 6= g ′ . I must verify that the result φ(g) or φ(g ′ ) is the same regardless of how I write the coset. (If φ(g) 6= φ(g ′ ) in this situation, then a single input — the coset ...
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... of (u) modulo m; if it is zero, (u) is said to be purely periodic^ modulo m. If all the terms of (u) after a certain point are divisible by m, so that the repeating part of (a) consists of the single residue zero, (u) is said to be a null sequence modulo m. Three important problems immediately sugge ...
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... vector field V on a smooth manifold M , we use subscripts for the point of evaluation: loosely, Vp ∈ Tp M , the tangent space to M at p. When V acts on f ∈ C ∞ (M ), we write the resulting smooth function simply as V f , so V f (g) = [Vg ](f ). For a smooth map F : M → N and p ∈ M , let D(F )p : Tp ...
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... Section 2 of this module is concerned with Demoivre’s theorem and its applications. We start in Subsection 2.1 by proving the theorem which states that (cos1θ + i1 sin1θ0)n = cos1(nθ0) + i1sin1(nθ0) (where i02 = −1), and then use it to derive trigonometric identities, in Subsection 2.2, and to find ...
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... Let (A, ·) be an associative algebra and R : A → A be a linear map satisfying R(x) · R(y) + R(x · y) = R(R(x) · y + x · R(y)), ∀x, y ∈ A. (13) Then x ∗ y = R(x) · y − y · R(x) − x · y, ∀x, y ∈ A defines a pre-Lie algebra. The above R is called Rota-Baxter map of weight 1. Generalization: Approach fr ...
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On Zero Semimodules of Systems over Semirings

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Homological algebra



Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.
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