Figure 8 - uploaded by George Mcnulty
Content may be subject to copyright.
Source publication
The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n+1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra
is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndo...
Context in source publication
Similar publications
This is a survey on stated skein algebras and their representations.
We give necessary and sufficient conditions, both algebraic and geometric, for a quadrilateral to be the level set of the sum of the distances to m ≥ 2 different lines.
This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of ( k , l ) -algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a ( k , l ) -algebraically st...
We present here a construction of an object which can be used as a free object for partial algebras and strong equations. Spectral algebras are useful in finding an algebraic characterization of classes of partial algebras definable by a set of strong equations.
Citations
... There has been little exploration of fine level membership for var-mem(A) within tractable cases. In [25], the authors show that var-mem(L) is in NL for Lyndon's algebra L. In [46] it is shown that a certain six-element semigroup AC 2 (also known as A g 2 [33], and not to be confused with the complexity class AC 2 ) has var-mem(AC 2 ) solvable in polynomial time. At the end of this section we show that their characterisation of the pseudovariety of AC 2 in fact leads to NL solvability of var-mem(AC 2 ). ...
We explore new interactions between finite model theory and a number of classical streams of universal algebra and semigroup theory. After refocussing some finite model theoretic tools in universal algebraic context, we present a number of results. A key result is an example of a finite algebra whose variety is not finitely axiomatisable in first order logic, but which has first order definable finite membership problem. This algebra witnesses the simultaneous failure of the Los-Tarski Theorem, the SP-preservation theorem and Birkhoff's HSP-preservation theorem at the finite level as well as providing a negative solution to the first order formulation of the long-standing Eilenberg Schützenberger problem. The example also shows that a pseudova-riety without any finite pseudo-identity basis may be finitely axiomatisable in first order logic. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra and a mapping from any fixed template constraint satisfaction problem to a first order equivalent variety membership problem, thereby providing examples of variety membership problems complete in each of the classes L, NL, Modp(L), P (provided they are nonempty), and infinitely many others (depending on complexity-theoretic assumptions).
... In Kun and Vértesi [18] it is shown that Θ(n k ) growth is possible (for any k ∈ N), while Kozik [16] showed that at least exponential growth is possible. McNulty, Szekely and Willard [22] give numerous concrete examples of algebras whose equational complexity is sandwiched somewhere between linear and quadratic growth, while Jackson and McNulty [13] give a linear growth rate for the equational complexity of Lyndon's algebra. ...
... In Jackson and McNulty [13] it is suggested that as well as finding high growth equational complexity, there should be equal interest in finding slow but unbounded growth. In particular, algebras of very slow, but unbounded growth, appear to be more likely related to difficult unresolved issues relating to axiomatizability. ...
The equational complexity function $\beta_\mathscr{V}:\mathbb{N}\to\mathbb{N}$ of an equational class of algebras $\mathscr{V}$ bounds the size of equation required to determine membership of $n$-element algebras in $\mathscr{V}$. Known examples of finitely generated varieties $\mathscr{V}$ with unbounded equational complexity have growth in $\Omega(n^c)$, usually for $c\geq \frac{1}{2}$. We show that much slower growth is possible, exhibiting $O(\log_2^3(n))$ growth amongst varieties of semilattice ordered inverse semigroups and additive idempotent semirings. We also examine a quasivariety analogue of equational complexity, and show that a finite group has polylogarithmic quasi-equational complexity function, bounded if and only if all Sylow subgroups are abelian.
... We wish to observe that A 1 2 either has non-polynomial bounded equational complexity, or has co-NP-easy membership problem (with respect to many-one reductions), or possibly both. The idea is on page 247 of Jackson and McNulty [20]: if the equational complexity of A 1 2 is bounded by a polynomial p(n), then given a finite semigroup B not in the variety of A 1 2 , we may guess an equation u ≈ v of length at most p(|B|) that is true on A 1 2 , as well as an assignment of its variables into B witnessing its failure on B. This certificate for non-membership in the variety of A 1 2 may be verified in deterministic polynomial time using [24,Lemma 8] to verify that u ≈ v holds on A 1 2 (as well as the easy check that the witnessing assignment into B does yield failure of u ≈ v on B). We believe this provides reasonable evidence that membership in the pseudovariety of A 1 2 is not NP-hard with respect to polynomial time many-one reductions. ...
We examine some flexible notions of constraint satisfaction, observing some
relationships between model theoretic notions of universal Horn class
membership and robust satisfiability. We show the \texttt{NP}-completeness of
$2$-robust monotone 1-in-3 3SAT in order to give very small examples of finite
algebras with \texttt{NP}-hard variety membership problem. In particular we
give a $3$-element algebra with this property, and solve a widely stated
problem by showing that the $6$-element Brandt monoid has \texttt{NP}-hard
variety membership problem. These are the smallest possible sizes for a general
algebra and a semigroup to exhibit \texttt{NP}-hardness for the membership
problem of finite algebras in finitely generated varieties.
