An axiomatisation of quantum logic

1973 ◽  
Vol 38 (3) ◽  
pp. 389-392 ◽  
Author(s):  
Ian D. Clark
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Ordered Set ◽  
Unary Operation ◽  
Axiom System ◽  
Partial Ordering ◽  
Orthomodular Poset ◽  
Implication Algebra ◽  
Image Position

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:(1) the greatest lower bound of a and Na exists and is 0,(2) a ≤ b implies Nb ≤ Na,(3) NNa = a.Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfyingwhere a ≤ b means a ∧ b = a, as usual.Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:An orthomodular lattice gives a Finch implication algebra by defining → byA Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ byThe orthocomplementation is unchanged in both cases.

scholarly journals Quantum logic as an implication algebra

1970 ◽  
Vol 2 (1) ◽  
pp. 101-106 ◽  
Author(s):  
P.D. Finch
Keyword(s):  
Boolean Algebra ◽  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Binary Operation ◽  
Unary Operation ◽  
Material Implication ◽  
Logical Implication ◽  
Partially Ordered ◽  
Implication Algebra

For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by “≤”, and which, as a poset, is orthocomplemented by a unary operation of negation. The negation of the proposition x is denoted by NX and the least element in the logic is denoted by 0, we write NO = 1.A binary operation “→” is introduced into a logic, the operation is interpreted as material implication so that “x → y” is a proposition of the logic and is read as “x materially implies y”. If material implication has the properties11. (x → 0) = NX, 12. if x ≤ y then (z → x) ≤ (z → y), 13. if x ≤ y then x ≤ (y ≤ z)= x → z, 14. x ≤ {y → N(y → Nx)}, then the logic is an orthomodular lattice. The lattice operations of join and meet are given by x ∨ y = Nx → N(Nx → Ny) x ∧ y = N(X → N(x → y)) and, in terms of the lattice operations, the material implication is given by (x → y) = (y ∧ x) ∨ NX.Moreover the logic is a Boolean algebra if, and only if, in addition to the properties above, material implication satifies 15. (x → y) = (Ny → Nx).

Injective Hulls of Semilattices

1970 ◽  
Vol 13 (1) ◽  
pp. 115-118 ◽  
Author(s):  
G. Bruns ◽  
H. Lakser
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Binary Operation ◽  
Ordered Set ◽  
Upper Bounds ◽  
Partial Ordering ◽  
Partially Ordered ◽  
Image Position ◽  
Injective Hulls

A (meet-) semilattice is an algebra with one binary operation ∧, which is associative, commutative and idempotent. Throughout this paper we are working in the category of semilattices. All categorical or general algebraic notions are to be understood in this category. In every semilattice S the relationdefines a partial ordering of S. The symbol "∨" denotes least upper bounds under this partial ordering. If it is not clear from the context in which partially ordered set a least upper bound is taken, we add this set as an index to the symbol; for example, ∨AX denotes the least upper bound of X in the partially ordered set A.

Free Involutorial Completely Simple Semigroups

1985 ◽  
Vol 37 (2) ◽  
pp. 271-295 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Maximal Subgroup ◽  
Universal Algebra ◽  
Simple Semigroup ◽  
Binary Operation ◽  
Unary Operation ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Image Position ◽  
Completely Simple

An involution x → x* of a semigroup S is an antiautomorphism of S of order at most 2, that is (xy)* = y*x* and x** = x for all x, y ∊ S. In such a case, S is called an involutorial semigroup if regarded as a universal algebra with the binary operation of multiplication and the unary operation *. If S is also a completely simple semigroup, regarded as an algebra with multiplication and the unary operation x → x−1 of inversion (x−1 is the inverse of x in the maximal subgroup of S containing x), then (S, −1, *), or simply S, is an involutorial completely simple semigroup. All such S form a variety determined by the identities above concerning * andwhere x0 = xx−1.

ANOTHER ARITHMETIC OF THE EVEN AND THE ODD

2018 ◽  
Vol 11 (3) ◽  
pp. 604-608
Author(s):  
CELIA SCHACHT
Keyword(s):  
Unary Operation ◽  
Axiom System ◽  
Integer Part ◽  
Image Position

AbstractThis article presents an axiom system for an arithmetic of the even and the odd, one that is stronger than those discussed in Pambuccian (2016) and Menn & Pambuccian (2016). It consists of universal sentences in a language extending the usual one with 0, 1, +, ·, <, – with the integer part of the half function $[{ \cdot \over 2}]$, and two unary operation symbols.

The consistency of the axiom of universality for the ordering of cardinalities

1985 ◽  
Vol 50 (2) ◽  
pp. 502-509
Author(s):  
Marco Forti ◽  
Furio Honsell
Keyword(s):  
Set Theory ◽  
Partially Ordered Set ◽  
Regular Cardinal ◽  
Ordered Set ◽  
Partial Ordering ◽  
Generic Extension ◽  
Ground Model ◽  
Partially Ordered ◽  
Transfer Method ◽  
Image Position

T. Jech [4] and M. Takahashi [7] proved that given any partial ordering R in a model of ZFC there is a symmetric submodel of a generic extension of where R is isomorphic to the injective ordering on a set of cardinals.The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of “cardinal universality” is consistent:CU. For any partially ordered set (X, ≼) there is a bijection f:X → Y such that(i.e. x ≼ y iff ∃g: f(x) → f(y) injective). (See [1].)The consistency of CU relative to ZF0 (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.In this paper the authors define a model of ZF + CU as a symmetric submodel of a generic extension obtained by forcing “à la Easton” with a class of conditions which add κ generic subsets to any regular cardinal κ of a ground model satisfying ZF + V = L.

scholarly journals Regular ω-semigroups

1968 ◽  
Vol 9 (1) ◽  
pp. 46-66 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Partial Ordering ◽  
Image Position

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe. If E = {ei: i∈ N}, where N denotes the set of all non-negative integers, and if the elements of E form the chainthen S is called an ω-semigroup.

scholarly journals REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

2017 ◽  
Vol 82 (2) ◽  
pp. 576-589 ◽  
Author(s):  
KOSTAS HATZIKIRIAKOU ◽  
STEPHEN G. SIMPSON
Keyword(s):  
Theorem Proving ◽  
Chain Condition ◽  
Reverse Mathematics ◽  
Partial Ordering ◽  
Young Diagrams ◽  
Equivalence Proof ◽  
Image Position ◽  
Countably Infinite ◽  
Infinite Set ◽  
Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

scholarly journals Classical Logic and Quantum Logic with Multiple and Common Lattice Models

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Mladen Pavičić
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Orthomodular Lattice ◽  
Lattice Models ◽  
Quantum Space ◽  
Technical Terms

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and nondistributive ortholattices. In particular, we prove that both classical logic and quantum logic are sound and complete with respect to each of these lattices. We also show that there is one common nonorthomodular lattice that is a model of both quantum and classical logic. In technical terms, that enables us to run the same classical logic on both a digital (standard, two-subset, 0-1-bit) computer and a nondigital (say, a six-subset) computer (with appropriate chips and circuits). With quantum logic, the same six-element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.

The Word Problem for Orthogroups

1981 ◽  
Vol 33 (4) ◽  
pp. 893-900 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Maximal Subgroup ◽  
Word Problem ◽  
Unary Operation ◽  
Group Inverse ◽  
Completely Regular Semigroup ◽  
Completely Regular ◽  
Image Position ◽  
Natural Way

A semigroup which is a union of groups is said to be completely regular. If in addition the idempotents form a subsemigroup, the semigroup is said to be orthodox and is called an orthogroup. A completely regular semigroup S is provided in a natural way with a unary operation of inverse by letting a-l for a ∈ S be the group inverse of a in the maximal subgroup of S to which a belongs. This unary operation satisfies the identities(1)(2)(3)In fact a completely regular semigroup can be defined as a unary semigroup (a semigroup with an added unary operation) satisfying these identities. An orthogroup can be characterized as a completely regular semigroup satisfying the additional identity(4)

scholarly journals Left residuated lattices induced by lattices with a unary operation

2019 ◽  
Vol 24 (2) ◽  
pp. 723-729
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Orthomodular Lattice ◽  
Residuated Lattice ◽  
Unary Operation ◽  
Residuated Lattices ◽  
Boolean Algebras ◽  
Natural Question ◽  
Similar Technique ◽  
Orthomodular Lattices

Abstract In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

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