Page 274 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 274
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
unchanged. By Theorem 4.4.13 the congruence structure is unchanged since both algebras share
the same maximal lattice image. £
In view of Corollary 7.1.12 and McKenzie [1975] Theorem 1.3, we also have:
Corollary 7.1.13. Any algebra A in a pointed ternary discriminator variety has a skew
Boolean ∩-algebra polynomial reduct whose congruences coincide with those of A.
For more on skew Boolean algebras and discriminator varieties, see the paper by Karin
Cvetko-Vah and Antonino Salibra in the following references. Also of interest is the paper by
Murskii which, among other things, shows that almost all finite algebras are discriminator
algebras (with “almost all” under stood in a certain sense).
References
J. C. Abbott,
Semi-Boolean algebras, Matematicki Vesnik 4 (1967), 177– 198
R. J. Bignall,
Quasiprimal varieties and components of universal algebras, Flinders University of
South Australia Dissertation,1976.
R. J. Bignall and J. Leech,
Skew Boolean algebras and discriminator varieties, A. Universalis 33 (1995), 387 – 398.
R. J. Bignall, and M. Spinks,
Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae
Japonicae, 58 (2003), 629 – 638.
Corrigendum: Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae
Mathematicae Japonicae, 66 (2007), 387–390.
S. Burris, S. and H. P. Sankappanavar,
A Course in Universal Algebra, Springer-Verlag, New York, 1981.
W. H. Cornish,
On Iseki’s BCK-algebras, Algebraic Structures and Applications, Lecture Notes in Pure
and Applied Mathematics, Vol. 74 (1982), Marcel Dekker, 101 – 122.
K. Cvetko-Vah and A. Salibra,
The connection of skew Boolean algebras and discriminator varieties to Church
algebras, Algebra Universalis, 73 (2015), 369-390.
K. Iseki and S. Tanaka,
An introduction to the theory of BCK-algebras, Math. Japonica 23 (1978), 1 – 26.
272
unchanged. By Theorem 4.4.13 the congruence structure is unchanged since both algebras share
the same maximal lattice image. £
In view of Corollary 7.1.12 and McKenzie [1975] Theorem 1.3, we also have:
Corollary 7.1.13. Any algebra A in a pointed ternary discriminator variety has a skew
Boolean ∩-algebra polynomial reduct whose congruences coincide with those of A.
For more on skew Boolean algebras and discriminator varieties, see the paper by Karin
Cvetko-Vah and Antonino Salibra in the following references. Also of interest is the paper by
Murskii which, among other things, shows that almost all finite algebras are discriminator
algebras (with “almost all” under stood in a certain sense).
References
J. C. Abbott,
Semi-Boolean algebras, Matematicki Vesnik 4 (1967), 177– 198
R. J. Bignall,
Quasiprimal varieties and components of universal algebras, Flinders University of
South Australia Dissertation,1976.
R. J. Bignall and J. Leech,
Skew Boolean algebras and discriminator varieties, A. Universalis 33 (1995), 387 – 398.
R. J. Bignall, and M. Spinks,
Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae Mathematicae
Japonicae, 58 (2003), 629 – 638.
Corrigendum: Implicative BCS-algebra subreducts of skew Boolean algebras, Scientiae
Mathematicae Japonicae, 66 (2007), 387–390.
S. Burris, S. and H. P. Sankappanavar,
A Course in Universal Algebra, Springer-Verlag, New York, 1981.
W. H. Cornish,
On Iseki’s BCK-algebras, Algebraic Structures and Applications, Lecture Notes in Pure
and Applied Mathematics, Vol. 74 (1982), Marcel Dekker, 101 – 122.
K. Cvetko-Vah and A. Salibra,
The connection of skew Boolean algebras and discriminator varieties to Church
algebras, Algebra Universalis, 73 (2015), 369-390.
K. Iseki and S. Tanaka,
An introduction to the theory of BCK-algebras, Math. Japonica 23 (1978), 1 – 26.
272
