Geodesic
On Semi-Boolean-Like Algebras
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 1, pp. 101-120 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In a previous paper, we introduced the notion of Boolean-like algebra as a generalisation of Boolean algebras to an arbitrary similarity type. In a nutshell, a double-pointed algebra $\mathbf {A}$ with constants $0,1$ is Boolean-like in case for all $a\in A$ the congruences $\theta \left( a,0\right) $ and $\theta \left( a,1\right) $ are complementary factor congruences of $\mathbf {A}$. We also introduced the weaker notion of semi-Boolean-like algebra, showing that it retained some of the strong algebraic properties characterising Boolean algebras. In this paper, we continue the investigation of semi-Boolean like algebras. In particular, we show that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional regular operations.
In a previous paper, we introduced the notion of Boolean-like algebra as a generalisation of Boolean algebras to an arbitrary similarity type. In a nutshell, a double-pointed algebra $\mathbf {A}$ with constants $0,1$ is Boolean-like in case for all $a\in A$ the congruences $\theta \left( a,0\right) $ and $\theta \left( a,1\right) $ are complementary factor congruences of $\mathbf {A}$. We also introduced the weaker notion of semi-Boolean-like algebra, showing that it retained some of the strong algebraic properties characterising Boolean algebras. In this paper, we continue the investigation of semi-Boolean like algebras. In particular, we show that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional regular operations.
Classification : 03C05, 06E75
Keywords: Boolean-like algebra; central element; noncommutative lattice theory 
@article{AUPO_2013_52_1_a8,
     author = {Ledda, Antonio and Paoli, Francesco and Salibra, Antonino},
     title = {On {Semi-Boolean-Like} {Algebras}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {101--120},
     year = {2013},
     volume = {52},
     number = {1},
     mrnumber = {3202753},
     zbl = {06285758},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2013_52_1_a8/}
}
TY  - JOUR
AU  - Ledda, Antonio
AU  - Paoli, Francesco
AU  - Salibra, Antonino
TI  - On Semi-Boolean-Like Algebras
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2013
SP  - 101
EP  - 120
VL  - 52
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/AUPO_2013_52_1_a8/
LA  - en
ID  - AUPO_2013_52_1_a8
ER  - 
%0 Journal Article
%A Ledda, Antonio
%A Paoli, Francesco
%A Salibra, Antonino
%T On Semi-Boolean-Like Algebras
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2013
%P 101-120
%V 52
%N 1
%U http://geodesic.mathdoc.fr/item/AUPO_2013_52_1_a8/
%G en
%F AUPO_2013_52_1_a8
Ledda, Antonio; Paoli, Francesco; Salibra, Antonino. On Semi-Boolean-Like Algebras. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 52 (2013) no. 1, pp. 101-120. http://geodesic.mathdoc.fr/item/AUPO_2013_52_1_a8/

[1] Bignall, R. J., Leech, J.: Skew Boolean algebras and discriminator varieties. Algebra Universalis 33 (1995), 387–398. | DOI | MR | Zbl

[2] Blok, W. J., Pigozzi, D.: On the structure of varieties with equationally definable principal congruences IV. Algebra Universalis 31 (1994), 1–35. | DOI | MR | Zbl

[3] Burris, S. N., Sankappanavar, H. P.: A Course in Universal Algebra. Springer, Berlin, 1981. | MR | Zbl

[4] Busaniche, M., Cignoli, R.: Constructive logic with strong negation as a substructural logic. Journal of Logic and Computation 20, 4 (2010), 761–793. | DOI | MR | Zbl

[5] Chajda, I., Halaš, R., Rosenberg, I. G.: Ideals and the binary discriminator in universal algebra. Algebra Universalis 42 (1999), 239–251. | DOI | MR | Zbl

[6] Comer, S.: Representations by algebras of sections over Boolean spaces. Pacific Journal of Mathematics 38 (1971), 29–38. | DOI | MR | Zbl

[7] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse on Substructural Logics. Elsevier, Amsterdam, 2007.

[8] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht, 1998. | MR | Zbl

[9] Jackson, M., Stokes, T.: Semigroups with if-then-else and halting programs. International Journal of Algebra and Computation 19, 7 (2009), 937–961. | DOI | MR | Zbl

[10] Koppelberg, S.: General theory of Boolean algebras. In: Koppelberg, S., Monk, J. D., Bonnet, R. (eds.): Handbook of Boolean Algebras, Vol. 1, North-Holland, Amsterdam, 1989. | MR

[11] Leech, J.: Skew lattices in rings. Algebra Universalis 26 (1989), 48–72. | DOI | MR | Zbl

[12] Leech, J.: Recent developments in the theory of skew lattices. Semigroup Forum 52 (1996), 7–24. | DOI | MR | Zbl

[13] Manzonetto, G., Salibra, A.: From $\lambda $-calculus to universal algebra and back. In: MFCS’08, volume 5162 of LNCS, (2008), 479–490. | MR | Zbl

[14] Paoli, F., Ledda, A., Kowalski, T., Spinks, M.: Quasi-discriminator varieties. (submitted).

[15] Salibra, A., Ledda, A., Paoli, F., Kowalski, T.: Boolean-like algebras. Algebra Universalis 69, 2 (2013), 113–138. | DOI | MR | Zbl

[16] Spinks, M.: On the Theory of Pre-BCK Algebras. PhD Thesis, Monash University, 2003.

[17] Vaggione, D.: Varieties in which the Pierce stalks are directly indecomposable. Journal of Algebra 184 (1996), 424–434. | DOI | MR | Zbl