TY - JOUR

T1 - Action of endomorphism semigroups on definable sets

AU - Mashevitzky, G.

AU - Plotkin, B.

AU - Plotkin, E.

N1 - Funding Information:
G. Mashevitzky owes much to the late Mati Rubin, whose friendly and professional support within years was of invaluable importance. Other authors share his feelings. We are grateful to B. Zilber for numerous discussions. The research of the third author was supported by ISF grants 1207/12, 1623/16 and the Emmy Noether Research Institute for Mathematics.
Publisher Copyright:
© 2018 World Scientific Publishing Company.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - The aim of the paper is to construct, discuss and apply the Galois-type correspondence between subsemigroups of the endomorphism semigroup End(A) of an algebra A and sets of logical formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of End(A) on definable sets over A. We treat some applications of this Galois correspondence. The first one concerns logic geometry. Namely, it gives a uniform approach to geometries defined by various fragments of the initial language. The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way, one can get an effective syntactical expression by semantic tools. Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first-order language and revealing new connections between well-known concepts. The fourth application concerns the generalization of the unification theory, or more generally Term Rewriting Theory, to the logic unification theory.

AB - The aim of the paper is to construct, discuss and apply the Galois-type correspondence between subsemigroups of the endomorphism semigroup End(A) of an algebra A and sets of logical formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of End(A) on definable sets over A. We treat some applications of this Galois correspondence. The first one concerns logic geometry. Namely, it gives a uniform approach to geometries defined by various fragments of the initial language. The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way, one can get an effective syntactical expression by semantic tools. Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first-order language and revealing new connections between well-known concepts. The fourth application concerns the generalization of the unification theory, or more generally Term Rewriting Theory, to the logic unification theory.

KW - Galois correspondence

KW - definable sets

KW - elementary embeddings

KW - endomorphisms

KW - homogeneous algebras

KW - semigroups of transformations

KW - unification types

UR - http://www.scopus.com/inward/record.url?scp=85052965225&partnerID=8YFLogxK

U2 - 10.1142/S0218196718400106

DO - 10.1142/S0218196718400106

M3 - Article

AN - SCOPUS:85052965225

SN - 0218-1967

VL - 28

SP - 1585

EP - 1612

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

IS - 8

ER -