From Structures

Contents 1 Involutive Monoids 1.1 Definition 1 1.2 Definition 2 1.3 Definition 3 1.4 Morphisms 1.5 Basic Results 1.6 Examples 1.7 Properties 1.8 Finite members 1.9 Subclasses 1.10 Superclasses

Involutive Monoids

InMon

Definition 1

An involutive monoid is a monoid with a unary operation of period two that antidistributes over the monoid operation, i.e. an algebra $\mathbf{A}=\langle A,\cdot,1,^\cup,\rangle$, where $\langle A,\cdot,1)$ is a monoid

${}^\cup$ has period two: $x^{\cup\cup} = x$

${}^\cup$ antidistributes over $\cdot$: $(x;y)^\cup = y^\cup;x^\cup$

Remark:

Definition 2

Definition 3

Morphisms

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

$h(xy)=h(x)h(y)$

Basic Results

$1={1^\cup}^\cup=(1^\cup;1)^\cup=1^\cup;{1^\cup}^\cup=1^\cup;1=1^\cup$

Examples

Properties

Classtype & variety
Equational theory & (un)decidable 
Quasiequational theory & (un)decidable
First-order theory & (un)decidable
Locally finite & yes/no
Residual size & n/...
Congruence distributive & yes/no
Congruence modular & yes/no
Congruence meet-semidistributive & yes/no
Congruence n-permutable & yes/no
Congruence regular & yes/no
Congruence uniform & yes/no
Congruence extension property & yes/no
Definable principal congruences & yes/no
Equationally def. pr. cong. & yes/no
Amalgamation property & yes/no
Strong amalgamation property & yes/no
Epimorphisms are surjective & yes/no

Finite members

$f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ \end{array}$

Subclasses

Superclasses