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Suslin Algebra
In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin. The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslin trees or Suslin lines. See also Andrei Suslin Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee ... References {{algebra-stub Boolean algebra Forcing (mathematics) Independence results ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Boolean Algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra ''A'' has an essentially unique completion, which is a complete Boolean algebra containing ''A'' such that every element is the supremum of some subset of ''A''. As a partially ordered set, this completion of ''A'' is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum. Examples Complete Boolean algebras *Every finite Boolean algebra is complete. *The algebra of subsets of a given set is a complete Boolean algebra. *The regular open sets of any topological space form a complete Boolean algebra. This example is of particular importance because every forcing poset can be considered as a topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom (order Theory)
In the mathematical field of order theory, an element ''a'' of a partially ordered set with least element 0 is an atom if 0 < ''a'' and there is no ''x'' such that 0 < ''x'' < ''a''. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0. Atomic orderings Let <: denote the in a partially ordered set. A partially ordered set with a least element 0 is atomic if every element ''b'' > 0 has an atom ''a'' below it, that is, there is some ''a'' such that ''b'' ≥ ''a'' :> ''0''. Every finite parti ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, Matrix (mathematics), matrices, Ring (mathematics), rings, and Field (mathematics), fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a Set (mathematics), set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Chain Condition
In order theory, a partially ordered set ''X'' is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in ''X'' is countable. Overview There are really two conditions: the ''upwards'' and ''downwards'' countable chain conditions. These are not equivalent. The countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound. This is called the "countable chain condition" rather than the more logical term "countable antichain condition" for historical reasons related to certain chains of open sets in topological spaces and chains in complete Boolean algebras, where chain conditions sometimes happen to be equivalent to antichain conditions. For example, if κ is a cardinal, then in a complete Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequence of elements, so chain conditions are equivalent to antichain conditions. Partial orders and s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Yakovlevich Suslin
Mikhail Yakovlevich Suslin (russian: Михаи́л Я́ковлевич Су́слин; , November 15, 1894 – 21 October 1919, Krasavka) (sometimes transliterated Souslin) was a Russian mathematician who made major contributions to the fields of general topology and descriptive set theory. Biography Mikhail Suslin was born on November 15, 1894 in the village of Krasavka, the only child of poor peasants Yakov Gavrilovich and Matrena Vasil'evna Suslin. From a young age, Suslin showed a keen interest in mathematics and was encouraged to continue his education by his primary school teacher, Vera Andreevna Teplogorskaya-Smirnova. From 1905 to 1913 he attended Balashov boys' grammar school. In 1913, Suslin enrolled at the Imperial Moscow University and studied under the tutelage of Nikolai Luzin. He graduated with a degree in mathematics in 1917 and immediately began working at the Ivanovo-Voznesensk Polytechnic Institute. Suslin died of typhus in the 1919 Moscow epidemic follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Tree
In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by ) or a Suslin algebra. The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ1) implies that there are no Suslin trees. More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain has cardinality less than κ. In particular a Suslin tree is the same as a ω1-Suslin tree. showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the Generalized Continuum Hypothesis implies the existence of an ℵ2-Suslin tree, is a longstanding open problem. See also * Glossary of set theory * Kurepa tree * List of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suslin Line
In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as , from the Cyrillic .) Formulation Suslin's problem asks: Given a non-empty totally ordered set ''R'' with the four properties # ''R'' does not have a least nor a greatest element; # the order on ''R'' is dense (between any two distinct elements there is another); # the order on ''R'' is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and # every collection of mutually disjoint non-empty open intervals in ''R'' is countable (this is the countable chain condition for the order topology of ''R''), is ''R'' necessarily order-isomorphic to the real line R? If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrei Suslin
Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee Chair and Professor of mathematics at Northwestern University. He was born on 27 December 1950 in St. Petersburg, Russia. As a youth, he was an "all Leningrad" gymnast. He received his PhD from Leningrad University in 1974; his thesis was titled ''Projective modules over polynomial rings''. In 1976 he and Daniel Quillen independently proved Serre's conjecture about the triviality of algebraic vector bundles on affine space. In 1982 he and Alexander Merkurjev proved the Merkurjev–Suslin theorem on the norm residue homomorphism in Milnor K2-theory, with applications to the Brauer group. Suslin was an invited speaker at the International Congress of Mathematicians in 1978 and 1994, and he gave a plenary invited address at the Congre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his '' An Investigation of the Laws of Thought'' (1854). According to Huntington, the term "Boolean algebra" wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. Forcing has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing. Intuition Intuitively, forcing consists of expanding the set theoretical universe V to a larger universe V^ . In this bigger universe, for example, one might have many new real numbers, identified with subsets of the set \mathbb of natural numbers, that were not there in the old ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
