On sentences which are true of direct unions of algebras

1951 ◽  
Vol 16 (1) ◽  
pp. 14-21 ◽  
Author(s):  
Alfred Horn
Keyword(s):  
Normal Form ◽  
Wide Class ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Factor Algebra ◽  
The Matrix ◽  
Distinct Individual ◽  
Direct Union ◽  
Individual Variables ◽  
Image Position

It is well known that certain sentences corresponding to similar algebras are invariant under direct union; that is, are true of the direct union when true of each factor algebra. An axiomatizable class of similar algebras, such as the class of groups, is closed under direct union when each of its axioms is invariant. In this paper we shall determine a wide class of invariant sentences. We shall also be concerned with determining sentences which are true of a direct union provided they are true of some factor algebra. In the case where all the factor algebras are the same, a further result is obtained. In §2 it will be shown that these criteria are the only ones of their kind. Lemma 7 below may be of some independent interest.We adopt the terminology and notation of McKinsey with the exception that the sign · will be used for conjunction. Expressions of the form ∼∊, where ∊ is an equation, will be called inequalities. In accordance with the analogy between conjunction and disjunction with product and sum respectively, we shall call α1, …, αn the terms of the disjunctionand the factors of the conjunctionEvery closed sentence is equivalent to a sentence in prenez normal form,where x1, …, xm distinct individual variables, Q1, …, Qm are quantifiers, and the matrix S is an open sentence in which each of the variables x1, …, xm actually occurs. The sentence S may be written in either disjunctive normal form:where αi,j is either an equation or an inequality, or in conjunctive normal form:.

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

On closure under direct product

1958 ◽  
Vol 23 (2) ◽  
pp. 149-154 ◽  
Author(s):  
C. C. Chang ◽  
Anne C. Morel
Keyword(s):  
Normal Form ◽  
Direct Product ◽  
Disjunctive Normal Form ◽  
Negative Answer ◽  
Sufficient Condition ◽  
Natural Question ◽  
Existential Quantifier ◽  
Relational Systems ◽  
Image Position

In 1951, Horn obtained a sufficient condition for an arithmetical class to be closed under direct product. A natural question which arose was whether Horn's condition is also necessary. We obtain a negative answer to that question.We shall discuss relational systems of the formwhere A and R are non-empty sets; each element of R is an ordered triple 〈a, b, c〉, with a, b, c ∈ A.1 If the triple 〈a, b, c〉 belongs to the relation R, we write R(a, b, c); if 〈a, b, c〉 ∉ R, we write (a, b, c). If x0, x1 and x2 are variables, then R(x0, x1, x2) and x0 = x1 are predicates. The expressions (x0, x1, x2) and x0 ≠ x1 will be referred to as negations of predicates.We speak of α1, …, αn as terms of the disjunction α1 ∨ … ∨ αn and as factors of the conjunction α1 ∧ … ∧ αn. A sentence (open, closed or neither) of the formwhere each Qi (if there be any) is either the universal or the existential quantifier and each αi, l is either a predicate or a negation of a predicate, is said to be in prenex disjunctive normal form.

On simplifying the matrix of a wff

1968 ◽  
Vol 33 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Peter Andrews
Keyword(s):  
Normal Form ◽  
Decision Procedure ◽  
Conjunctive Normal Form ◽  
Predicate Calculus ◽  
First Order ◽  
The Matrix ◽  
Order Predicate Calculus

In [3], [4], and [5] Joyce Friedman formulated and investigated certain rules which constitute a semi-decision procedure for wffs of first order predicate calculus in closed prenex normal form with prefixes of the form ∀x1 … ∀xκ∃y1 … ∃ym∀z1 … ∀zn. Given such a wff QM, where Q is the prefix and M is the matrix in conjunctive normal form, Friedman's rules can be used, in effect, to construct a matrix M* which is obtained from M by deleting certain conjuncts of M.

scholarly journals On a Normal Form of the Orthogonal Transformation I

1958 ◽  
Vol 1 (1) ◽  
pp. 31-39 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Quadratic Form ◽  
Lorentz Transformation ◽  
Orthogonal Transformation ◽  
Complex Conjugate ◽  
Linear Transformations ◽  
Characteristic Roots ◽  
The Matrix ◽  
Real Characteristic ◽  
Image Position

At the Edmonton Meeting of the Canadian Mathematical Congress E. Wigner asked me whether one knew something about the distribution of the characteristic roots of the linear transformations that leave invariant the quadratic form t2+x2-y2-z2, just as one knows that a Lorentz transformation has two complex conjugate characteristic roots and two real characteristic roots that are either inverse to one another or the numbers 1 and -1.In this paper an answer to E. Wigner’s question will be obtained.We are concerned with the pairs of matrices (X,A) with coefficients in a field of reference F such that the condition0.1is satisfied, where XT = (ξki) is the transpose of the matrix X = (ξki). It follows that both matrices are quadratic of the same degree d.

scholarly journals Interval Valued Fuzzy Soft Sets and Fuzzy Connectives

2019 ◽  
Vol 8 (4) ◽  
pp. 20-23
Keyword(s):  
Normal Form ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Soft Set ◽  
Fuzzy Subset ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Fuzzy Soft Sets ◽  
Interval Valued ◽  
Fuzzy Connectives

The Article, We Learning A Few Operations Of Interval Valued Fuzzy Soft Sets Of Connectives And Give Elementary Properties Of Interval Valued Fuzzy Soft Sets Of Principal Disjunctive Normal Form And Principal Conjunctive Normal Form. 2000 Ams Subject Classification: 03f55, 08a72, 20n25. Keywords: Interval Valued Fuzzy Subset, Interval Valued Fuzzy Soft Set, And Principal Conjunctive Normal Form And Principal Disjunctive Normal Form Interval Valued Fuzzy Soft Set ‘’∧ ‘’ Operator And ‘’∨’’ Operator.

COMMUTATIVE IMPLICATIONS ON COMPLETE LATTICES

1994 ◽  
Vol 02 (03) ◽  
pp. 333-341 ◽  
Author(s):  
WANGMING WU
Keyword(s):  
Normal Form ◽  
Complete Lattice ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Complete Lattices

This paper is devoted to the investigation of commutative implications on a complete lattice L. It is proved that the disjunctive normal form (DNF) of a linguistic composition * is included in the conjunctive normal form (CNF) of that *, i.e., DNF(*) ≤ CNF(*) holds, for a special family of t-norms, t-conorms and negations induced by commutative implications.

The decision problem for formulas with a small number of atomic subformulas

1973 ◽  
Vol 38 (3) ◽  
pp. 471-480 ◽  
Author(s):  
Harry R. Lewis ◽  
Warren D. Goldfarb
Keyword(s):  
Normal Form ◽  
Decision Problem ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Principal Result ◽  
The Other ◽  
Universal Quantifiers

In this paper we consider classes of quantificational formulas specified by restrictions on the number of atomic subformulas appearing in a formula. Little seems to be known about the decision problem for such classes, except that the class whose members contain at most two distinct atomic subformulas is decidable [2]. (We use “decidable” and “undecidable” throughout with respect to satisfiability rather than validity. All undecidable problems to which we refer are of maximal r.e. degree.) The principal result of this paper is the undecidability of the class of those formulas containing five atomic subformulas and with prefixes of the form ∀∃∀…∀. In fact, we show the undecidability of two sub-classes of this class: one (Theorem 1) consists of formulas whose matrices are in disjunctive normal form with two disjuncts; the other (Corollary 1) consists of formulas whose matrices are in conjunctive normal form with three conjuncts. (Theorem 1 sharpens Orevkov's result [8] that the class of formulas in disjunctive normal form with two disjuncts is undecidable.) A second corollary of Theorem 1 shows the undecidability of the class of formulas with prefixes of the form ∀…∀∃, containing six atomic subformulas, and in conjunctive normal form with three conjuncts. These restrictions to prefixes ∀∃∀…∀ and ∀…∀∃ are optimal. For by a result of the first author [5], any class of prenex formulas obtained by restricting both the number of atomic formulas and the number of universal quantifiers is reducible to a finite class of formulas, and so each such class is decidable; and the class of formulas with prefixes ∃…∃∀…∀ is, of course, decidable.

scholarly journals SYMMETRIES IN MODAL LOGICS

2015 ◽  
Vol 21 (4) ◽  
pp. 373-401
Author(s):  
CARLOS ARECES ◽  
EZEQUIEL ORBE
Keyword(s):  
Normal Form ◽  
Wide Class ◽  
Conjunctive Normal Form ◽  
Modal Logics ◽  
Symmetry Detection ◽  
Detection Problem ◽  
Graph Automorphism ◽  
Theoretical Foundations

AbstractIn this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [5]. Hence, the results apply to a wide class of modal logics including, for example, hybrid logics. We present two graph constructions that enable the reduction of symmetry detection in modal formulas to the graph automorphism detection problem, and we evaluate the graph constructions on modal benchmarks.

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