generalized_bl-algebras - MathStructures

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=====Generalized BL-algebras=====

Abbreviation: **GBL**

====Definition====
A \emph{generalized BL-algebra} is a [[residuated lattice]] 
$\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$x\wedge y=y\cdot(y\backslash x\wedge e)$, $x\wedge y=(x/y\wedge e)\cdot y$

==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be generalized BL-algebras. A
morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$
that is a homomorphism: 

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash
y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  |undecidable |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |no |
^[[Congruence e-regular]]  |yes |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |

====Finite members====
^$n$       | 1 | 2 | 3 | 4 |  5 |  6 |  7 |   8 | 9 | 10 | 11 |
^# of algs | 1 | 1 | 2 | 5 | 10 | 23 | 49 | 111 |  |   |   |   
^# of si's | 1 | 1 | 2 | 4 |  9 | 19 | 42 |  97 |  |   |   |



====Subclasses====
[[Generalized MV-algebras]] 

[[Basic logic algebras]] 


====Superclasses====
[[Distributive residuated lattices]] 


====References====

[(Ln19xx>
)]

Trace: • relation_algebras • sqrt-quasi-mv-algebras • congruence_distributive • generalized_effect_algebras • principal_ideal_domains • start

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