### Table of Contents

## Semilattices with identity

Abbreviation: **Slat$_1$**

### Definition

A \emph{semilattice with identity} is a structure $\mathbf{S}=\langle S,\cdot,1\rangle$ of type $\langle 2,0\rangle $ such that

$\langle S,\cdot\rangle$ is a semilattices

$1$ is an indentity for $\cdot$: $x\cdot 1=x$

##### Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices with identity. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

### Examples

Example 1:

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | decidable in PTIME |

Quasiequational theory | decidable |

First-order theory | undecidable |

Locally finite | yes |

Residual size | 2 |

Congruence distributive | no |

Congruence modular | no |

Congruence meet-semidistributive | yes |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | yes |

Definable principal congruences | yes |

Equationally def. pr. cong. | no |

Amalgamation property | yes |

Strong amalgamation property | yes |

Epimorphisms are surjective | yes |

### Finite members

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

f(2)= &1\quad

f(3)= &1\quad

f(4)= &2\quad

f(5)= &5\quad

f(6)= &15

\end{array}$