## 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:

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

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

Classtype variety no unbounded yes yes yes, $n=2$ yes, $e=1$

### Finite members

 $n$ # of algs # of si's 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 1 1 3 3 8 12 35 61 167 0 1 1 2 3 7 12 31 59 161 329 944 2067 6148 14558 44483 116372

### Superclasses

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

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