### Table of Contents

## m-zeroids

Abbreviation: **MZrd**

### Definition

An \emph{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

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

### 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 | 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 |

### Subclasses

### Superclasses

### References

J. B. Palmatier and F. Guzman, \emph{M-zeroids structure and categorical equivalence}, Studia Logica, \textbf{100}(5) 2012, 975–1000