The finite basis problem for additively idempotent semirings of order four, II

The finite basis problem for semigroups asks when a given finite semigroup has a finite basis for its identities. This problem is one of the most investigated in variety theory. In this note we look at some easily established equivalent decision problems.

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, partial algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown to be equivalent to the finite identity basis problem for the finite members of a finiteiy based variety with definable principal congruences.

We establish a criterion for a semigroup identity to hold in the monoid of n times n upper unitriangular matrices with entries in a commutative semiring S. This criterion is combinatorial modulo the arithmetic of the multiplicative identity element of S. In the case where S is non-trivial and idempotent, the generated variety is the variety mathbf {J}_{mathbf {n-1}}, which by a result of Volkov is generated by any one of: the monoid of unitriangular Boolean matrices, the monoid R_n of all reflexive relations on an n element set, or the Catalan monoid C_n. We propose S-matrix analogues of these latter two monoids in the case where S is an idempotent semiring whose multiplicative identity element is the ‘top’ element with respect to the natural partial order on S, and show that each generates mathbf {J}_{mathbf {n-1}}. As a consequence we obtain a complete solution to the finite basis problem for Lossy gossip monoids.

The finite dimensional basis problem with an appendix on nets of Grassmann manifolds

On the Stabilization Problem for Submodules of Specht Modules

5 Finite basis problem

We exhibit finite algebras each generating a variety with NP-complete finite algebra membership problem. The smallest of these algebras is the flat graph algebra belonging to the tetrahedral graph, a graph of 6 vertices obtained by cutting and spreading out the surface of a tetrahedron on the plane. The sequence of graphs we use to build up our flat graph algebras is similar to the sequence exhibited by Wheeler in [36] , 1979, to describe the first order theory of k-colorable graphs. Graph algebras were introduced by Shallon in [34] , 1979, and investigated, among others, by Baker, McNulty and Werner in [2] , 1987. Flat algebras were constructed and used by McKenzie in [27] , 1996, to settle some open questions related to decidability, like Tarski's Finite Basis Problem. Flat graph algebras were also discussed by Willard in [37] , 1996, and Delić in [8] , 1998.

The finite basis problem in the pseudovariety joins of aperiodic semigroups with groups

The finite basis problem

Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

In terms of their approach to creative work, mathematicians display a spectrum of tendencies. Some focus most of their time and effort on building up a monumental theory. Sophus Lie was such a mathematician, with his focus on his theory of transformation groups. Among Frobenius’ mentors, Weierstrass, with his focus on the theory of abelian integrals and functions and the requisite foundations in complex function theory, and Richard Dedekind, with his theory of algebraic numbers and ideals, are further examples of mathematicians who were primarily theory builders. At the other end of the spectrum are mathematicians whose focus was first and foremost on concrete mathematical problems. Of course, many mathematicians fall somewhere between these extremes. A prime example is Hilbert, who created several far-reaching theories, such as his theory of integral equations, but also solved many specific problems, such as the finite basis problem in the theory of invariants, Waring’s problem, and Dirichlet’s problem; and of course he posed his famous 23 mathematical problems for others to attempt to solve. Frobenius was decidedly at the problem-solver end of the spectrum. Virtually all of his important mathematical achievements were driven by the desire to solve specific mathematical problems, not famous long-standing problems such as Waring’s problem, but in general, problems that he perceived in the mathematics of his time.

R. McKenzie has recently associated to each Turing machine T \mathcal {T} a finite algebra A ( T ) \mathbf {A} (\mathcal {T}) having some remarkable properties. We add to the list of properties, by proving that the equational theory of A ( T ) \mathbf {A}(\mathcal {T}) is finitely axiomatizable if T \mathcal {T} halts on the empty input. This completes an alternate (and simpler) proof of McKenzie’s negative answer to A. Tarski’s finite basis problem. It also removes the possibility, raised by McKenzie, of using A ( T ) \mathbf {A}(\mathcal {T}) to answer an old question of B. Jónsson.

In these lectures we return to the RS Problem discussed in E. Kiss’s article (this volume). We discuss the current status of the problem and describe some results which arose from the study of the problem, including the solution to Tarski’s finite basis problem. A subtitle for these lectures might be “Some recent results in general algebra, mostly due to R. McKenzie.”

Noetherianity and Specht problem for varieties of bicommutative algebras

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.