FL$_c$-algebras

Abbreviation: FL$_c$

Definition

A FL$_c$-algebra is an FL-algebra $\mathbf{A}=\langle A, \vee, \wedge, \cdot, 1, \backslash, /, 0\rangle$ such that

$\cdot$ is contractive: $x\le x\cdot x$

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

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: $h(x ... y)=h(x) ... h(y)$

Definition

An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype (value, see description)
Equational theory undecidable1)
Quasiequational theory undecidable
First-order theory undecidable
Locally finite no
Residual size infinite
Congruence distributive yes
Congruence modular yes
Congruence $n$-permutable yes
Congruence regular
Congruence uniform
Congruence extension property
Definable principal congruences
Equationally def. pr. cong.
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

Superclasses

References


1) K. Chvalovsky and R. Horcík, Full Lambek calculus with contraction is undecidable, Journal of Symbolic Logic, 81(2), 524–540