Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

On group automorphisms in universal algebraic geometry

In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.

ALGEBRAIC GEOMETRY FOR MV-ALGEBRAS

In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

Automorphic equivalence of linear algebras

This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈ Θ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras H1 and H2 are called geometrically equivalent if and only if the H1-closed sets coincide with the H2-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H1 and H2 are called automorphically equivalent if and only if the H1-closed sets coincide with the H2-closed sets after some "changing of coordinates". We can detect the difference between geometric and automorphic equivalence of algebras of the variety Θ by researching of the automorphisms of the category Θ0 of the finitely generated free algebras of the variety Θ. By [5] the automorphic equivalence of algebras provided by inner automorphism coincide with the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group 𝔄/𝔜 is big. Here 𝔄 is the group of all automorphisms of the category Θ0, 𝔜 is a normal subgroup of all inner automorphisms of the category Θ0. In [6] the variety of all Lie algebras and the variety of all associative algebras over the infinite field k were studied. If the field k has not nontrivial automorphisms then group 𝔄/𝔜 in the first case is trivial and in the second case has order 2. We consider in this paper the variety of all linear algebras over the infinite field k. We prove that group 𝔄/𝔜 is isomorphic to the group (U(kS2)/U(k{e}))λ Aut k, where S2 is the symmetric group of the set which has 2 elements, U(kS2) is the group of all invertible elements of the group algebra kS2, e ∈ S2, U(k{e}) is a group of all invertible elements of the subalgebra k{e}, Aut k is the group of all automorphisms of the field k. So even the field k has not nontrivial automorphisms the group 𝔄/𝔜 is infinite. This kind of result is obtained for the first time. The example of two linear algebras which are automorphically equivalent but not geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.