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m-zeroid [2016/11/21 10:29]
jipsen created
m-zeroid [2016/11/21 11:09] (current)
jipsen
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====Basic results==== ====Basic results====
 +All subdirectly irreducible algebras are linearly ordered.
 +
 +The lattice is always bounded, with top element $0$.
 +
 +The bottom element $-0$ is the identity of $+$.
 +
 +The dual operation $x\cdot y=-(-y+-x)$ is the fusion of a commutative integral involutive semilinear residuated lattice. In fact, m-zeroids are precisely the duals of these residuated lattices, which are also known as involutive IMTL algebras.
====Properties==== ====Properties====
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^$n$       | 1 | 2 | 3 | 4 | 5 | 6 |  7 |  8 |  9 |  10 |  11 |  12 |   13 |   14 |    15 |    16 |     17 | ^$n$       | 1 | 2 | 3 | 4 | 5 | 6 |  7 |  8 |  9 |  10 |  11 |  12 |   13 |   14 |    15 |    16 |     17 |
-^# of algs | 1 | 1 | 1 | 3 |  |  |   |   |   |    |     |     |      |      |       |       |        |+^# of algs | 1 | 1 | 1 | 3 | 3 | 8 | 12 | 35 | 61 | 167 |     |     |      |      |       |       |        |
^# of si's | 0 | 1 | 1 | 2 | 3 | 7 | 12 | 31 | 59 | 161 | 329 | 944 | 2067 | 6148 | 14558 | 44483 | 116372 | ^# of si's | 0 | 1 | 1 | 2 | 3 | 7 | 12 | 31 | 59 | 161 | 329 | 944 | 2067 | 6148 | 14558 | 44483 | 116372 |
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====Subclasses==== ====Subclasses====
-[[]] +[[TBD]]
====Superclasses==== ====Superclasses====
-[[]]  +[[TBD]]
- +
-[[]]  +
- +
-[[]] +
====References==== ====References====
-[(PG1994> 
J. B. Palmatier and F. Guzman, J. B. Palmatier and F. Guzman,
\emph{M-zeroids structure and categorical equivalence}, \emph{M-zeroids structure and categorical equivalence},
Studia Logica, Studia Logica,
-\textbf{100}(5) 2012, 975--1000)] +\textbf{100}(5) 2012, 975--1000
- +
-[(COM2000> +
-Roberto L. O. Cignoli, Itala M. L. D'Ottaviano, Daniele Mundici, +
-\emph{Algebraic foundations of many-valued reasoning}, +
-Trends in Logic---Studia Logica Library +
-\textbf{7} Kluwer Academic Publishers +
-2000, x+231)] +
- +
-[(Mu1987> +
-Daniele Mundici, +
-\emph{Bounded commutative BCK-algebras have the amalgamation property}, +
-Math. Japon., +
-\textbf{32} 1987, 279--282)]+