Generic algorithms decide problems on sets of almost all inputs, outputting an indefinite answer for other rare inputs. We will prove that the word problem is generically decidable in finitely generated semigroups 𝔖, for which there exists a congruence θ such that the semigroup 𝔖/ θ is an infinite residually finite monoid with cancellation property and decidable word problem. This generalizes the author’ earlier result on generic decidability of the word problem in finitely presented semigroups that remain infinite when adding commutativity and cancelling properties. Examples of such semigroups are one-relator semigroups as well as so-called balanced semigroups, for which generic decidability of the word problem has been proved by Won. In particular, balanced are Tseitin and Makanin’s classical semigroups with undecidable word problem.
The joint logic of problems and propositions QHC introduced by S. A. Melikhov, as well as intuitionistic modal logic QH4, is studied. An immersion of these logics into classical first-order predicate logic is considered. An analog of the Löwenheim–Skolem theorem on the existence of countable elementary submodels for QHC and QH4 is established.
It is proved that a singular superalgebra with a 2-dimensional even part is isomorphic to a superalgebra B 2|3 (φ , ξ, ψ ) . In particular, there do not exist infinite-dimensional simple singular superalgebras with a 2-dimensional even part. It is proved that if a singular superalgebra contains an odd left annihilator, then it contains a nondegenerate switch. Lastly, it is established that for any number N ≥ 5, except the numbers 6, 7, 8, 11, there exist singular superalgebras with a switch of dimension N . For the numbers N = 6, 7, 8, 11, there do not exist singular N -dimensional superalgebras with a switch.
The concept of P-stability is a particular case of generalized stability of complete theories. We study injective S -acts with a P -stable theory. It is proved that the class of injective S -acts is ( P , 1)-stable only if S is a one-element monoid. Also we describe commutative and linearly ordered monoids S the class of injective S -acts over which is ( P , s )-, ( P , a )-, and ( P , e )-stable.
We consider a class of generalized derivations that arise in connection with the problem of adjoining unity to an algebra with generalized derivation, and of searching envelopes for Novikov–Poisson algebras. Conditions for the existence of the localization of an algebra with ternary derivation are specified, as well as conditions under which given an algebra with ternary derivation, we can construct a Novikov–Poisson algebra and a Jordan superalgebra. Finally, we show how the simplicity of an algebra with Brešar generalized derivation is connected with simplicity of the appropriate Novikov algebra.
