## The structure of pseudocomplemented distributive lattices. II. Congruence extension and amalgamation

HTML articles powered by AMS MathViewer

- by G. Grätzer and H. Lakser PDF
- Trans. Amer. Math. Soc.
**156**(1971), 343-358 Request permission

## Abstract:

This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes ${\mathcal {B}_n}, - 1 \leqq n \leqq \omega$, which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (i.e., the semigroup generated by the operators**, and**

*H, S***) are described. The Amalgamation Property is shown to hold iff $n \leqq 2$ or $n = \omega$. For $3 \leqq n < \omega ,{\mathcal {B}_n}$ does not satisfy the Amalgamation Property; the deviation is measured by a class Amal $({\mathcal {B}_n})( \subseteq {\mathcal {B}_n})$. The finite algebras in Amal $({\mathcal {B}_n})$ are determined.**

*P*## References

- A. H. Clifford and G. B. Preston,
*The algebraic theory of semigroups. Vol. II*, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR**0218472** - P. M. Cohn,
*Universal algebra*, Harper & Row, Publishers, New York-London, 1965. MR**0175948**
S. D. Comer and J. S. Johnson, - Alan Day,
*Injectives in non-distributive equational classes of lattices are trivial*, Arch. Math. (Basel)**21**(1970), 113–115. MR**274357**, DOI 10.1007/BF01220888 - George Grätzer,
*Universal algebra*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR**0248066** - Bjarni Jónsson,
*Extensions of relational structures*, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 146–157. MR**0202601** - Bjarni Jónsson,
*Sublattices of a free lattice*, Canadian J. Math.**13**(1961), 256–264. MR**123493**, DOI 10.4153/CJM-1961-021-0 - H. Lakser,
*The structure of pseudocomplemented distributive lattices. I. Subdirect decomposition*, Trans. Amer. Math. Soc.**156**(1971), 335–342. MR**274358**, DOI 10.1090/S0002-9947-1971-0274358-7 - K. B. Lee,
*Equational classes of distributive pseudo-complemented lattices*, Canadian J. Math.**22**(1970), 881–891. MR**265240**, DOI 10.4153/CJM-1970-101-4
D. Pigozzi,

*The standard semigroup of operators of a variety*(manuscript).

*On some operations on classes of algebras*, Notices Amer. Math. Soc.

**13**(1966), 829. Abstract #639-1.

## Additional Information

- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**156**(1971), 343-358 - MSC: Primary 06.50
- DOI: https://doi.org/10.1090/S0002-9947-1971-0274359-9
- MathSciNet review: 0274359