Open Questions

It is an open question whether the uniform word problem for the class of all meadows is solvable. In particular, it is not known whether the equational axiomatization of meadows admits Knuth-Bendix completion.

The equational theory of the class of all meadows is decidable. However, it is an open question whether the equational theory of the meadow of rational numbers is decidable.

Bergstra and Tucker (2007) have shown that there exists a finite equational specification, without hidden functions, of the meadow of rational numbers under initial algebra semantics. It is an open question whether there exists a finite equational specification, without hidden functions, of the meadow of rational numbers under initial algebra semantics that constitutes a complete term rewriting system.

Bergstra and Middelburg (2011) have shown that a finite equational specification, without hidden functions, of the meadow of rational numbers under initial algebra semantics can be obtained by adding the equation (1 + x² + y²) · (1 + x² + y²)-1 = 1 to the equational axiomatization of meadows. It is an open question whether a finite equational specification, without hidden functions, of the meadow of rational numbers under initial algebra semantics can be obtained by adding a finite set of equations in one variable to the equational axiomatization of meadows.

An open question related to the previous one is the question whether there exists a logically weakest set of equations among the finite sets of equations whose addition to the equational axiomatization of meadows yields an equational specification, without hidden functions, of the meadow of rational numbers under initial algebra semantics.

It is also an open question whether the equational theory of the class of all meadows that satisfy the above-mentioned equation (1 + x² + y²) · (1 + x² + y²)-1 = 1 is decidable. This question differs from the question whether the equational theory of the meadow of rational numbers is decidable because the former equational theory happens to be a proper subset of the latter equational theory.

In Bergstra, Bethke, and Ponse (2013), expansions of meadows with certain operations, such as the sign operation and the floor and ceiling operations, are presented. It is an open question whether the particular equational axioms for these operations, in combinations with the equational axiomatization of meadows, are independent.

Bergstra, Bethke, and Ponse (2015) have shown that there exists an infinite equational axiomatization of the meadow of complex numbers expanded with imaginary unit and complex conjugation. It is an open question whether there exists a finite equational axiomatization of the meadow of complex numbers expanded with imaginary unit and complex conjugation.

The multiplicative inverse operation of a meadow is an involution. In Bergstra and Middelburg (2015a), non-involutive meadows are introduced as variants of meadows in which multiplicative inverse of zero is different from zero (but not an additional value). The class of all non-involutive meadows constitutes a variety. It is an open question whether the class of all algebras that are either (involutive) meadows or non-involutive meadows constitutes a variety.

Divisive meadows are variants of meadows in which the multiplicative inverse operation is replaced by a division operation. Bergstra and Middelburg (2015b) have shown that, in each divisive meadow of prime characteristic for which there exists an n such that each element of its carrier is the root of a non-trivial polynomial of degree n or less, it holds that each term is equal to a simple fraction. It is an open question whether, in each divisive meadow of non-zero characteristic for which there exists an n such that each element of its carrier is the root of a non-trivial polynomial of degree n or less, it holds that each term is equal to a simple fraction.

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