Abstract
In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of Sheffer stroke and a Hilbert algebra. After it is shown that the axioms of Sheffer stroke Hilbert algebra are independent, it is given some properties of this algebraic structure. Then it is stated the relationship between Sheffer stroke Hilbert algebra and Hilbert algebra by defining a unary operation on Sheffer stroke Hilbert algebra. Also, it is presented deductive system and ideal of this algebraic structure. It is defined an ideal generated by a subset of a Sheffer stroke Hilbert algebra, and it is constructed a new ideal of this algebra by adding an element of this algebra to its ideal.
Highlights
Sheffer stroke was initially introduced by H.M
We have introduced a Sheffer Stroke Hilbert algebra, and studied deductive systems and ideals on these algebras
We showed that a Sheffer Stroke Hilbert algebra is a Hilbert algebra, where the binary operation −→ is defined by x −→ y := x|(y|y)
Summary
Sheffer stroke was initially introduced by H.M. Sheffer [13]. This operation drew many researchers’ attention since any Boolean operation or function is expressed via only this binary operation [10] This leads to reductions of axioms or formulas for many algebraic structures. Skolem for studies in intuitionistic and other nonclassical logics [5] These algebraic structures can be thought as parts of the propositional logic including the implication and the distinguished element 1. Diego studied Hilbert algebras, their deductive systems and various properties. We introduce a Sheffer Stroke Hilbert algebra and describe its a deductive systems and ideals. We prove that a Sheffer Stroke Hilbert algebra is a Hilbert algebra where x −→ y := x|(y|y) After that, it is introduced a unary operation ∗ on this structure. We construct an ideal of a Sheffer stroke Hilbert algebra by adding a new element
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