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Abstract: It is proved that 2-torsion-free simple right-symmetric superrings having a nontrivial idempotent and satisfying a superidentity (x, y, z) + (−1)z(x+y)(z, x, y) + (−1)x(y+z)(y, z, x) = 0 are associative. As a consequence, every simple finitedimensional (1, 1)-superalgebra with semisimple even part over an algebraically closed field of characteristic 0 is associative. PubDate: 2021-09-23

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Abstract: Let G be a permutation group of a set Ω and k be a positive integer. The k-closure of G is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(Ω) which has the same orbits as has G under the componentwise action on the set Ωk. It is proved that the k-closure of a finite nilpotent group coincides with the direct product of k-closures of all of its Sylow subgroups. PubDate: 2021-09-20

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Abstract: We continue to study into the notion of general recursive realizability, which was introduced in [Algebra and Logic, 59, No. 5, 367-384 (2020)], based on using indices of general recursive functions as a constructive way of obtaining some realizations from others. It is proved that intuitionistic logic is not sound with respect to a weaker version of the semantics of general recursive realizability. PubDate: 2021-09-20

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Abstract: Let N be a quasivariety of torsion-free nilpotent groups of class at most two. It is proved that the set of subquasivarieties in N, which have no independent basis of quasiidentities and are generated by a finitely generated group, is infinite. It is stated that there exists an infinite set of quasivarieties M in N which are generated by a finitely generated group and are such that for every quasivariety K(M ⊈ K ⊆ N), an interval [M, K] has the power of the continuum in the quasivariety lattice. PubDate: 2021-09-20

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Abstract: We look at specific features of the algebraic structure of an upper semilattice of computable families of computably enumerable sets in Ω. It is proved that ideals of minuend and finite families of Ω coincide. We deal with the question whether there exist atoms and coatoms in the factor semilattice of Ω with respect to an ideal of finite families. Also we point out a sufficient condition for computable families to be complemented. PubDate: 2021-09-20

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Abstract: We introduce the notion of a Σω-bounded structure and specify a necessary and sufficient condition for a universal Σ-function to exist in a hereditarily finite superstructure over such a structure, for the class of all unary partial Σ-functions assuming values in the set ω of natural ordinals. Trees and equivalences are exemplified in hereditarily finite superstructures over which there exists no universal Σ-function for the class of all unary partial Σ-functions, but there exists a universal Σ-function for the class of all unary partial Σ-functions assuming values in the set ω of natural ordinals. We construct a tree T of height 5 such that the hereditarily finite superstructure ℍ(T) over T has no universal Σ-function for the class of all unary partial Σ-functions assuming values 0, 1 only. PubDate: 2021-09-20

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Abstract: We describe coordinate groups of generalized rigid metabelian groups in which, whenever a group is noncommutative, the second factor of a rigid series is a divisible R-module over an appropriate integral domain R. PubDate: 2021-09-20

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Abstract: Let K be an algebraically closed field of characteristic 0, and let G be a divisible ordered Abelian group. Maclane [Bull. Am. Math. Soc., 45, 888-890 (1939)] showed that the Hahn field K((G)) is algebraically closed. Our goal is to bound the lengths of roots of a polynomial p(x) over K((G)) in terms of the lengths of its coefficients. The main result of the paper says that if ð›¾ is a limit ordinal greater than the lengths of all of the coefficients, then the roots all have length less than ωωð›¾. PubDate: 2021-09-20

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Abstract: In a first order language we interpret the action of the monoid M of embeddings of (ℚ,<) on the set ℚ inside (M,°). A similar result is proved for the monoid E of all endomorphisms of (ℚ,≤). PubDate: 2021-06-17 DOI: 10.1007/s10469-021-09628-w

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Abstract: We consider different characterizations of computability by means of infinite time Blum–Shub–Smale machines (ITBM) via specific functions on sets and computable infinitary formulas. PubDate: 2021-06-15 DOI: 10.1007/s10469-021-09625-z

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Abstract: On the set of all first-order theories T(σ) of similarity type σ, a binary operation {·} is defined by the rule T · S = Th({A × B A = T and B = S}) for any theories T,S ∈ T(σ). The structure 〈T(σ); ⋅〉 forms a commutative semigroup, which is called a semigroup of theories. We prove that a semigroup of theories is an ideal extension of a semigroup \( {S}_T^{\ast } \) by a semigroup ST . The set of all idempotent elements of a semigroup of theories forms a complete lattice with respect to the partial order ≤ defined as T ≤ S iff T · S = S for all T,S ∈ T(σ). Also the set of all idempotent complete theories forms a complete lattice with respect to ≤, which is not necessarily a sublattice of the lattice of idempotent theories. PubDate: 2021-06-15 DOI: 10.1007/s10469-021-09623-1

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Abstract: Unary functions definable in o-stable ordered groups of nonvaluational type are studied. Such functions are proved to be piecewise monotone and continuous. PubDate: 2021-06-15 DOI: 10.1007/s10469-021-09624-0

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Abstract: We describe topological properties, ranks, and closures as well as their dynamics for families of theories. Closures for families of theories are introduced based on sentences of the given theories. We look at properties of these closures and establish their relationship with standard closures. Values of e1-spectra, as well as conditions for the existence of least generating sets, are characterized. Closures for linearly ordered families of theories are examined. PubDate: 2021-06-12 DOI: 10.1007/s10469-021-09626-y

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Abstract: A basis for the commutator subgroup of a partially commutative metabelian pro-p-group is described. PubDate: 2021-06-12 DOI: 10.1007/s10469-021-09627-x

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Abstract: We consider a φ-logic L(ω) of a frame of order type ω endowed with an irreflexive operator. The irreflexive modality in LC was treated by the author in [Sib. Math. J., 55, No. 1, 185-190 (2014)] where it was shown that this modality on the class of finite chains, on the one hand, and on a single chain of order type ω, on the other hand, generates inconsistent φ-logics over LC. There, also, it was stated that L(ω) defines a new nonconstant connective in LC. Here we establish that the φ-logic L(ω) is Novikov complete over LC. PubDate: 2021-01-01 DOI: 10.1007/s10469-021-09618-y

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Abstract: We describe topological properties, ranks, closures, and their dynamics for families of theories. Types of topologies for families of theories are characterized. A relationship is established between ranks and topologies for families of theories. Boolean combinations of s-definable families of theories are treated, ranks and degrees with respect to these families are found, and values of the characteristics in question are described. We study closures of families of theories with respect to s-definable subfamilies and their Boolean combinations, properties of closure operators, and also a condition for the existence of a least generating set. Rank values for families of theories are specified in terms of algebras of definable subfamilies. PubDate: 2021-01-01 DOI: 10.1007/s10469-021-09620-4

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Abstract: We deal with subsemigroups of a free left regular band Fn of rank n. It is proved that any finite, right hereditary, left regular band with a linearly ordered support semilattice is embedded in Fn for some n. PubDate: 2021-01-01 DOI: 10.1007/s10469-021-09619-x