arXiv subjects

Olga Sapir

Publications and source records attributed to Olga Sapir.

7 recordsLinked to original sources

arXiv2020

Finitely based sets of 2-limited block-2-simple words

Let $\mathfrak A$ be an alphabet and $W$ be a set of words in the free monoid ${\mathfrak A}^*$. Let $S(W)$ denote the Rees quotient over the ideal of ${\mathfrak A}^*$ consisting of all words that are not subwords of words in $W$. A set of words $W$ is called {\em finitely based} if the monoid $S(W)$ is finitely based. A word $\bf u$ is called 2-limited if each variable occurs in $\bf u$ at most twice. A {\em block} of a word $\bf u$ is a maximal subword of $\bf u$ that does not contain any linear variables. We say that a word $\bf u$ is {\em block-2-simple} if each block of $\bf u$ involves at most two distinct variables. We provide an algorithm that recognizes finitely based sets of words among sets of 2-limited block-2-simple words. We also present new sufficient conditions under which a set of words is non-finitely based.

math.GR↗

arXiv2018

Lee monoid $L_4^1$ is non-finitely based

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new proof to the results of Zhang-Luo and Lee that the semigroup $L_\ell$ is non-finitely based each $\ell \ge 3$.

math.GR↗

arXiv2018

Lee monoids are non-finitely based while the sets of their isoterms are finitely based

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We show that every monoid which generates a variety containing $L_5^1$ and is contained in the variety generated by $L_\ell^1$ for some $\ell \ge 5$ is non-finitely based. We establish this result by analyzing $τ$-terms for $M$ where $τ$ is certain non-trivial congruence on the free semigroup, that is, we analyze words $\bf u$ with the property that ${\bf u} τ{\bf v}$ whenever $M$ satisfies an identity ${\bf u} \approx {\bf v}$. We also show that if $τ$ is the trivial congruence on the free semigroup and $\ell \le 5$ then the $τ$-terms (isoterms) for $L_\ell^1$ carry no information about the non-finite basis property of $L_\ell^1$.

math.GR↗

arXiv2016

The finite basis problem for words with at most two non-linear variables

Let A be an alphabet and W be a set of words in the free monoid A*. Let S(W) denote the Rees quotient over the ideal of A* consisting of all words that are not subwords of words in W. We call a set of words W finitely based if the monoid S(W) is finitely based. We find a simple algorithm that recognizes finitely based words among words with at most two non-linear variables. We also describe syntactically all hereditary finitely based monoids of the form S(W).

math.GR↗

arXiv2016

The finite basis problem for the monoid of 2 by 2 upper triangular tropical matrices

For each positive $n$, let $u_n = v_n$ denote the identity obtained from the Adjan identity $(xy) (yx) (xy) (xy) (yx) = (xy) (yx) (yx) (xy) (yx)$ by substituting $(xy) \rightarrow (x_1 x_2 \dots x_n)$ and $(yx) \rightarrow (x_n \dots x_2 x_1)$. We show that every monoid which satisfies $u_n = v_n$ for each positive $n$ and generates the variety containing the bicyclic monoid is nonfinitely based. This implies that the monoid of 2 by 2 upper triangular tropical matrices over the tropical semiring is nonfinitely based.

math.GR↗

arXiv2015

Finitely based monoids

We present a method for proving that a semigroup is finitely based and find some new sufficient conditions under which a monoid is finitely based. As an application, we find a class of finite monoids where the finite basis property behaves in a complicated way with respect to the lattice operations but can be recognized by a simple algorithm. The method results in a short proof of the theorem of E. Lee that every monoid that satisfies xtxysy = xtyxsy and xytxsy = yxtxsy is finitely based. Also, the method gives an alternative proof of the theorem of F. Blanchet-Sadri that a pseudovariety of n-testable languages is finitely based if and only if n < 4.

math.GR↗

arXiv2015

Non-finitely based monoids

We present a general method for proving that a semigroup is non-finitely based. The method is strong enough to cover the non-finite basis arguments in articles [1,3,4,5,7,8, 11,14,16,21,27,31,36,37]. In particular, the method allows to generalize the results in [1,8,36,37] and to simplify their proofs. The method also allows to remove one of the requirements on the "special system of identities" used by P. Perkins in [16] to find the first two examples of finite non-finitely based semigroups. We use our method to prove eleven new sufficient conditions under which a monoid is non-finitely based. As an application, we find infinitely many new examples of finite finitely based aperiodic monoids whose direct product is non-finitely based.

math.GR↗