Two Axiomatizations of Nelson Algebras

Adam Grabowski. "Two Axiomatizations of Nelson Algebras." Formalized Mathematics 23.2 (2015): 115-125. <http://eudml.org/doc/271778>.

@article{AdamGrabowski2015,
abstract = {Nelson algebras were first studied by Rasiowa and Białynicki- Birula [1] under the name N-lattices or quasi-pseudo-Boolean algebras. Later, in investigations by Monteiro and Brignole [3, 4], and [2] the name “Nelson algebras” was adopted - which is now commonly used to show the correspondence with Nelson’s paper [14] on constructive logic with strong negation. By a Nelson algebra we mean an abstract algebra 〈L, T, -, ¬, →, ⇒, ⊔, ⊓〉 where L is the carrier, − is a quasi-complementation (Rasiowa used the sign ~, but in Mizar “−” should be used to follow the approach described in [12] and [10]), ¬ is a weak pseudo-complementation → is weak relative pseudocomplementation and ⇒ is implicative operation. ⊔ and ⊓ are ordinary lattice binary operations of supremum and infimum. In this article we give the definition and basic properties of these algebras according to [16] and [15]. We start with preliminary section on quasi-Boolean algebras (i.e. de Morgan bounded lattices). Later we give the axioms in the form of Mizar adjectives with names corresponding with those in [15]. As our main result we give two axiomatizations (non-equational and equational) and the full formal proof of their equivalence. The second set of equations is rather long but it shows the logical essence of Nelson lattices. This formalization aims at the construction of algebraic model of rough sets [9] in our future submissions. Section 4 contains all items from Th. 1.2 and 1.3 (and the itemization is given in the text). In the fifth section we provide full formal proof of Th. 2.1 p. 75 [16].},
author = {Adam Grabowski},
journal = {Formalized Mathematics},
keywords = {quasi-pseudo-Boolean algebras; Nelson lattices; de Morgan lattices; De Morgan lattices; formalization},
language = {eng},
number = {2},
pages = {115-125},
title = {Two Axiomatizations of Nelson Algebras},
url = {http://eudml.org/doc/271778},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Adam Grabowski
TI - Two Axiomatizations of Nelson Algebras
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 2
SP - 115
EP - 125
AB - Nelson algebras were first studied by Rasiowa and Białynicki- Birula [1] under the name N-lattices or quasi-pseudo-Boolean algebras. Later, in investigations by Monteiro and Brignole [3, 4], and [2] the name “Nelson algebras” was adopted - which is now commonly used to show the correspondence with Nelson’s paper [14] on constructive logic with strong negation. By a Nelson algebra we mean an abstract algebra 〈L, T, -, ¬, →, ⇒, ⊔, ⊓〉 where L is the carrier, − is a quasi-complementation (Rasiowa used the sign ~, but in Mizar “−” should be used to follow the approach described in [12] and [10]), ¬ is a weak pseudo-complementation → is weak relative pseudocomplementation and ⇒ is implicative operation. ⊔ and ⊓ are ordinary lattice binary operations of supremum and infimum. In this article we give the definition and basic properties of these algebras according to [16] and [15]. We start with preliminary section on quasi-Boolean algebras (i.e. de Morgan bounded lattices). Later we give the axioms in the form of Mizar adjectives with names corresponding with those in [15]. As our main result we give two axiomatizations (non-equational and equational) and the full formal proof of their equivalence. The second set of equations is rather long but it shows the logical essence of Nelson lattices. This formalization aims at the construction of algebraic model of rough sets [9] in our future submissions. Section 4 contains all items from Th. 1.2 and 1.3 (and the itemization is given in the text). In the fifth section we provide full formal proof of Th. 2.1 p. 75 [16].
LA - eng
KW - quasi-pseudo-Boolean algebras; Nelson lattices; de Morgan lattices; De Morgan lattices; formalization
UR - http://eudml.org/doc/271778
ER -