By R. Padmanabhan
The significance of equational axioms emerged in the beginning with the axiomatic method of Boolean algebras, teams, and jewelry, and later in lattices. This exact study monograph systematically offers minimum equational axiom-systems for varied lattice-related algebras, whether they're given by way of subscribe to and meet or different forms of operations resembling ternary operations. all the axiom-systems is coded in a convenient manner in order that possible stick to the ordinary connection one of the numerous axioms and to appreciate tips to mix them to shape new axiom structures.
a brand new subject during this booklet is the characterization of Boolean algebras in the classification of all uniquely complemented lattices. right here, the distinguished challenge of E V Huntington is addressed, which -- in line with G Gratzer, a number one professional in glossy lattice idea -- is likely one of the difficulties that formed a century of analysis in lattice conception. between different issues, it truly is proven that there are infinitely many non-modular lattice identities that strength a uniquely complemented lattice to be Boolean, therefore offering a number of new axiom structures for Boolean algebras in the category of all uniquely complemented lattices. ultimately, a number of comparable strains of study are sketched, within the kind of appendices, together with one via Dr Willian McCune of the collage of latest Mexico, on functions of recent theorem-proving to the equational concept of lattices
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Additional info for Axioms For Lattices And Boolean Algebras
X ∨ (y ∧ (z ∧ x)) = x. Take y := z, z := y ∧ (z ∧ x) in 44, then use 35 with x := z ∧ x. 47. (x ∧ y) ∨ y = y. Identity 45 reduces to 47 by 34 with x := x ∧ y. 48. x ∨ (y ∧ (x ∧ z)) = x. Take z := x ∧ z in 46, then use 42 with y := z. 49. ((x ∧ y) ∨ (z ∧ y)) ∨ y = y. Take x := ((x ∧ y) ∨ (z ∧ y)) ∨ (((x ∧ y) ∨ (z ∧ y)) ∧ u) in 47, then use 41. 50. ((x ∧ y) ∨ (y ∧ z)) ∨ y = y. Take z := y ∧ z in 49, then use 42 with x := y, y := z. ∧ ∨ ∨ Finally note that identities 28, 40, 48 and 50 are L∧ 24 , L25 , L24 and L25 , respectively.
2 below provides a very simple identity to show that any finitely-based variety of algebras that admits a J´ onsson term and is definable by absorption identities, is one-based. 2 stated there without proof. 3. (Padmanabhan ) Let T be an equational theory with a majority polynomial p. For arbitrary polynomials f and g, the validity of two identities f = x and g = x in T is equivalent to p(f, g, y) = x, where y is a variable not occurring in f or g. Proof: Clearly f = x and g = x together imply p(f, g, y) = x.
4 Defining Lattices by Other Tools AxiomLattices 37 where < a, b, c > is not defined if the second equality in (28) does not hold, and which satisfies certain axioms. Conversely, his axioms imply only a weaker form of (28), namely (a ∧ b) ∨ (b ∧ c) ∨ (c ∧ a) ≤ < a, b, c > ≤ (a ∨ b) ∧ (b ∨ c) ∧ (c ∨ a) . Katri˜ na ´k  has shown that two operations < > and < > satisfying the axioms need not coincide even if they have the same domain of definition. Other problems concern self-dual varieties of lattices.
Axioms For Lattices And Boolean Algebras by R. Padmanabhan