Absolute subretracts and weak injectives in congruence modular varieties

Authors:
Brian A. Davey and L. G. Kovács

Journal:
Trans. Amer. Math. Soc. **297** (1986), 181-196

MSC:
Primary 08B30; Secondary 08B10, 16A52, 20E10

DOI:
https://doi.org/10.1090/S0002-9947-1986-0849474-X

MathSciNet review:
849474

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Abstract: Absolute subretracts and weak injectives in congruence modular varieties of universal algebras are investigated by focusing attention on the directly indecomposibles. The proofs rely on a congruence modular version of generalized direct products (direct products with amalgamation) and on the generalized Jónsson Lemma for congruence modular varieties. The results have immediate application to varieties of groups or rings.

**[1]**B. A. Davey and H. Werner,*Injectivity and Boolean powers*, Math. Z.**166**(1979), 205-223. MR**526465 (80b:08004)****[2]**B. A. Davey, K. R. Miles and V. J. Schumann,*Quasi-identities, Mal'cev conditions and congruence regularity*, preprint, La Trobe Univ., 1986. MR**911557 (89a:08001)****[3]**R. Freese,*On Jónsson's theorem*, Algebra Univ.**18**(1984), 70-76. MR**743457 (86e:08007)****[4]**R. Freese and R. McKenzie,*Residually small varieties with modular congruence lattices*, Trans. Amer. Math. Soc.**264**(1981), 419-430. MR**603772 (83d:08012a)****[5]**-,*The commutator, an overview*, preprint, Univ. of Hawaii, 1979.**[6]**O. C. Garcia and F. Larrión,*Injectivity in varieties of groups*, Algebra Univ.**14**(1982), 280-286. MR**654396 (83f:20021a)****[7]**G. Grätzer and H. Lakser,*The structure of pseudocomplemented distributive lattices*. III. Injectives and absolute subretracts, Tran. Amer. Math. Soc.**169**(1972), 475-487. MR**0309821 (46:8926)****[8]**H. P. Gumm,*An easy way to the commutator in modular varieties*, Arch, der Math.**34**(1980), 220-228. MR**590312 (81m:08015)****[9]**-,*Geometrical methods in congruence modular varieties*, Mem. Amer. Math. Soc.**45**No. 286 (1983).**[10]**J. Hagemann and C. Herrmann,*A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity*, Arch. Math.**32**(1979), 234-245. MR**541622 (80j:08006)****[11]**C. Herrmann,*Affine algebras in congruence modular varieties*, Acta Sci. Math. (Szeged)**41**(1979), 119-125. MR**534504 (80h:08011)****[12]**L. G. Kovács and M. F. Newman,*Injectives in varieties of groups*, Algebra Univ.**14**(1982), 398-400. MR**654406 (83f:20021b)****[13]**H. Neumann,*Varieties of groups*, Ergebnisse Math. Grenzgeb., Band 37, Springer-Verlag, Berlin, Heidelberg and New York, 1967. MR**0215899 (35:6734)****[14]**W. Taylor,*Some applications of the term condition*, Algebra Univ.**14**(1982), 11-24. MR**634412 (83d:08004)****[15]**H. Werner,*Congruences on products of algebras and functionally complete algebras*, Algebra Univ.**4**(1974), 99-105. MR**0360416 (50:12866)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0849474-X

Keywords:
Injectivity,
congruence modularity,
varieties of universal algebras,
groups,
rings

Article copyright:
© Copyright 1986
American Mathematical Society