On automorphism groups of relational systems and universal algebras

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

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Application of Higher-Order Ultraproducts to the Theory of Local Properties in Universal Algebras and Relational Systems

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-27.4.617 ◽

1973 ◽

Vol s3-27 (4) ◽

pp. 617-637 ◽

Cited By ~ 2

Author(s):

W. G. Malcolm

Keyword(s):

Higher Order ◽

Universal Algebras ◽

Relational Systems ◽

Local Properties

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Automorphism Groups of Unary Algebras on Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-127-2 ◽

1969 ◽

Vol 21 ◽

pp. 1165-1171 ◽

Cited By ~ 3

Author(s):

G. H. Wenzel

Keyword(s):

Automorphism Group ◽

Wreath Product ◽

Systematic Study ◽

Automorphism Groups ◽

Unary Algebra ◽

Main Result Theorem ◽

Universal Algebras ◽

Result Theorem ◽

The Right ◽

Characterization Theorems

This paper presents a systematic study of the automorphism groups of those unary (universal) algebras whose carrier set G is the carrier set of some group ) and whose automorphism set contains the right translations of the latter group. These algebras appear, apart from the known classical contexts, repeatedly in characterization theorems of endomorphism semigroups (End) and automorphism groups (Aut) of algebras due to Grätzer (3; 4; 5), Makkai (7), Armbrust and Schmidt (1), Birkhoff (2), and others.Our main result (Theorem 1) constitutes an essential strengthening of a theorem of Birkhoff and represents the automorphism group of a unary algebra (where F is contained in the set of left translations of the group as wreath product of two groups that are easily determined from F and G.

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