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m-zeroids

Abbreviation: MZrd

Definition

An m-zeroid is a algebra $\mathbf{A}=\langle A, \wedge, \vee, +, 0, -\rangle$ such that

$\langle A, +\rangle$ is a commutative semigroup

$\langle A, \wedge, \vee\rangle$ is a lattice

$-x=x$

$x + 0 = 0$

$x + -x = 0$

$x\le y\iff 0=-x+y$

$x + (y\vee z) = (x+y)\vee(x+z)$

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be MV-algebras. A morphism from $\mathbf{A} $ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\vee y)=h(x)\vee h(y)$, $h(-x)=-h(x)$, $h(0)=0$

Examples

Example 1:

Basic results

Properties

Finite members

$n$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
# of algs 1 1 1 3
# of si's 0 1 1 2 3 7 12 31 59 161 329 944 2067 6148 14558 44483 116372

see http://oeis.org/A030453

Subclasses

References