On semi-boolean-like algebras

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.

[1]  W. Blok,et al.  On the structure of varieties with equationally definable principal congruences IV , 1994 .

[2]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[3]  Stanley Burris,et al.  A course in universal algebra , 1981, Graduate texts in mathematics.

[4]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[5]  Marcel Jackson,et al.  SEMIGROUPS WITH if-then-else AND HALTING PROGRAMS , 2009, Int. J. Algebra Comput..

[6]  Jonathan Leech,et al.  Skew lattices in rings , 1989 .

[7]  D. Vaggione.  Varieties in Which the Pierce Stalks Are Directly Indecomposable , 1996 .

[8]  Francesco Paoli,et al.  Quasi-discriminator varieties , 2014, Int. J. Algebra Comput..

[9]  Manuela Busaniche,et al.  Constructive Logic with Strong Negation as a Substructural Logic , 2010, J. Log. Comput..

[10]  Francesco Paoli,et al.  Boolean-like algebras , 2013 .

[11]  Jonathan Leech,et al.  Skew Boolean algebras and discriminator varieties , 1995 .

[12]  George Metcalfe.  A SEQUENT CALCULUS FOR CONSTRUCTIVE LOGIC WITH STRONG NEGATION AS A SUBSTRUCTURAL LOGIC , 2009 .

[13]  Jonathan Leech,et al.  Recent developments in the theory of skew lattices , 1996 .

[14]  Giulio Manzonetto,et al.  From lambda-Calculus to Universal Algebra and Back , 2008, MFCS.

[15]  Giulio Manzonetto,et al.  From λ-calculus to Universal Algebra and Back , 2008 .

[16]  I. Chajda,et al.  Ideals and the binary discriminator in universal algebra , 1999 .

[17]  S. D. Comer,et al.  Representations by algebras of sections over Boolean spaces. , 1971 .