Extension of semi-lattices and lattices

Extension of meet semi-lattices

[1]:

from galactic.algebras.lattice import FiniteMeetSemiLattice
from galactic.algebras.poset import HasseDiagram
from galactic.algebras.examples.arithmetic import PrimeFactorsRenderer
from galactic.algebras.examples.arithmetic import PrimeFactors

[2]:

meet_semi_lattice = FiniteMeetSemiLattice[PrimeFactors](
    domain=[
        PrimeFactors(3),
        PrimeFactors(7),
        PrimeFactors(2 * 7),
        PrimeFactors(3 * 5),
    ]
)
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[2]:

../../_images/notebooks_lattice_extensions_2_0.png

[3]:

meet_semi_lattice.extend([PrimeFactors(3 * 7)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[3]:

../../_images/notebooks_lattice_extensions_3_0.png

[4]:

meet_semi_lattice.extend([PrimeFactors(2 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[4]:

../../_images/notebooks_lattice_extensions_4_0.png

[5]:

meet_semi_lattice.extend([PrimeFactors(11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[5]:

../../_images/notebooks_lattice_extensions_5_0.png

[6]:

meet_semi_lattice.extend([PrimeFactors(3 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[6]:

../../_images/notebooks_lattice_extensions_6_0.png

[7]:

meet_semi_lattice.extend([PrimeFactors(2 * 7 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[7]:

../../_images/notebooks_lattice_extensions_7_0.png

[8]:

meet_semi_lattice.extend([PrimeFactors(2 * 2 * 3 * 5 * 7 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[8]:

../../_images/notebooks_lattice_extensions_8_0.png

[9]:

meet_semi_lattice.extend([PrimeFactors(2 * 3 * 7 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[9]:

../../_images/notebooks_lattice_extensions_9_0.png

[10]:

meet_semi_lattice.extend([PrimeFactors(3 * 7 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[10]:

../../_images/notebooks_lattice_extensions_10_0.png

[11]:

meet_semi_lattice.extend([PrimeFactors(5)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[11]:

../../_images/notebooks_lattice_extensions_11_0.png

[12]:

meet_semi_lattice.extend([PrimeFactors(3 * 5 * 13)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[12]:

../../_images/notebooks_lattice_extensions_12_0.png

[13]:

meet_semi_lattice.extend([PrimeFactors(2 * 5)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[13]:

../../_images/notebooks_lattice_extensions_13_0.png

[14]:

meet_semi_lattice.extend([PrimeFactors(35)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[14]:

../../_images/notebooks_lattice_extensions_14_0.png

[15]:

meet_semi_lattice.extend([PrimeFactors(2 * 3 * 3 * 5 * 7 * 11)])
HasseDiagram[PrimeFactors](
    meet_semi_lattice, domain_renderer=PrimeFactorsRenderer(meet_semi_lattice=True)
)

[15]:

../../_images/notebooks_lattice_extensions_15_0.png

Extension of join semi-lattices

[16]:

from galactic.algebras.lattice import FiniteJoinSemiLattice
from galactic.algebras.poset import HasseDiagram
from galactic.algebras.examples.arithmetic import PrimeFactorsRenderer
from galactic.algebras.examples.arithmetic import PrimeFactors

[17]:

join_semi_lattice = FiniteJoinSemiLattice[PrimeFactors](
    domain=[
        PrimeFactors(3),
        PrimeFactors(7),
        PrimeFactors(2 * 7),
        PrimeFactors(3 * 5),
    ]
)
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[17]:

../../_images/notebooks_lattice_extensions_18_0.png

[18]:

join_semi_lattice.extend([PrimeFactors(2 * 11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[18]:

../../_images/notebooks_lattice_extensions_19_0.png

[19]:

join_semi_lattice.extend([PrimeFactors(11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[19]:

../../_images/notebooks_lattice_extensions_20_0.png

[20]:

join_semi_lattice.extend([PrimeFactors(3 * 11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[20]:

../../_images/notebooks_lattice_extensions_21_0.png

[21]:

join_semi_lattice.extend([PrimeFactors(2 * 7 * 11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[21]:

../../_images/notebooks_lattice_extensions_22_0.png

[22]:

join_semi_lattice.extend([PrimeFactors(2 * 2 * 3 * 5 * 7 * 11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[22]:

../../_images/notebooks_lattice_extensions_23_0.png

[23]:

join_semi_lattice.extend([PrimeFactors(3 * 7 * 11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[23]:

../../_images/notebooks_lattice_extensions_24_0.png

[24]:

join_semi_lattice.extend([PrimeFactors(5)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[24]:

../../_images/notebooks_lattice_extensions_25_0.png

[25]:

join_semi_lattice.extend([PrimeFactors(3 * 5 * 13)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[25]:

../../_images/notebooks_lattice_extensions_26_0.png

[26]:

join_semi_lattice.extend([PrimeFactors(2 * 5)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[26]:

../../_images/notebooks_lattice_extensions_27_0.png

[27]:

join_semi_lattice.extend([PrimeFactors(35)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[27]:

../../_images/notebooks_lattice_extensions_28_0.png

[28]:

join_semi_lattice.extend([PrimeFactors(2 * 3 * 3 * 5 * 7 * 11)])
HasseDiagram[PrimeFactors](
    join_semi_lattice, domain_renderer=PrimeFactorsRenderer(join_semi_lattice=True)
)

[28]:

../../_images/notebooks_lattice_extensions_29_0.png

Extension of lattices

First example

[29]:

from galactic.algebras.lattice import FiniteLattice
from galactic.algebras.poset import HasseDiagram
from galactic.algebras.examples.arithmetic import PrimeFactorsRenderer
from galactic.algebras.examples.arithmetic import PrimeFactors

[30]:

lattice = FiniteLattice[PrimeFactors](
    domain=[
        PrimeFactors(3),
        PrimeFactors(7),
        PrimeFactors(2 * 7),
        PrimeFactors(3 * 5),
    ]
)
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[30]:

../../_images/notebooks_lattice_extensions_32_0.png

[31]:

lattice.extend([PrimeFactors(3 * 7)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[31]:

../../_images/notebooks_lattice_extensions_33_0.png

[32]:

lattice.extend([PrimeFactors(2 * 11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[32]:

../../_images/notebooks_lattice_extensions_34_0.png

[33]:

lattice.extend([PrimeFactors(11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[33]:

../../_images/notebooks_lattice_extensions_35_0.png

[34]:

lattice.extend([PrimeFactors(3 * 11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[34]:

../../_images/notebooks_lattice_extensions_36_0.png

[35]:

lattice.extend([PrimeFactors(2 * 7 * 11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[35]:

../../_images/notebooks_lattice_extensions_37_0.png

[36]:

lattice.extend([PrimeFactors(2 * 2 * 3 * 5 * 7 * 11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[36]:

../../_images/notebooks_lattice_extensions_38_0.png

[37]:

lattice.extend([PrimeFactors(3 * 7 * 11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[37]:

../../_images/notebooks_lattice_extensions_39_0.png

[38]:

lattice.extend([PrimeFactors(5)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[38]:

../../_images/notebooks_lattice_extensions_40_0.png

[39]:

lattice.extend([PrimeFactors(3 * 5 * 13)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[39]:

../../_images/notebooks_lattice_extensions_41_0.png

[40]:

lattice.extend([PrimeFactors(2 * 5)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[40]:

../../_images/notebooks_lattice_extensions_42_0.png

[41]:

lattice.extend([PrimeFactors(5 * 7)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[41]:

../../_images/notebooks_lattice_extensions_43_0.png

[42]:

lattice.extend([PrimeFactors(2 * 3 * 3 * 5 * 7 * 11)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[42]:

../../_images/notebooks_lattice_extensions_44_0.png

[43]:

from galactic.algebras.lattice import ReducedContextDiagram

ReducedContextDiagram(
    lattice,
    domain_renderer=PrimeFactorsRenderer(meet_irreducible=True),
    co_domain_renderer=PrimeFactorsRenderer(join_irreducible=True),
)

[43]:

../../_images/notebooks_lattice_extensions_45_0.png

[44]:

from galactic.algebras.relational import BinaryTable

BinaryTable(
    lattice.reduced_context,
    domain_renderer=PrimeFactorsRenderer(),
    co_domain_renderer=PrimeFactorsRenderer(meet_irreducible=True),
)

[44]:

	60060	13860	90090	4290	15015	2730	462	770
\(11=11\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)
\(7=7\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(2=2\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(3=3\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(5=5\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)
\(195=3·5·13\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(4620=2^2·3·5·7·11\)	\(\checkmark\)	\(\checkmark\)
\(6930=2·3^2·5·7·11\)		\(\checkmark\)	\(\checkmark\)

Second example

[45]:

lattice = FiniteLattice[PrimeFactors](domain=[PrimeFactors(1)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[45]:

../../_images/notebooks_lattice_extensions_48_0.png

[46]:

lattice.extend([PrimeFactors(2)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[46]:

../../_images/notebooks_lattice_extensions_49_0.png

[47]:

lattice.extend([PrimeFactors(12)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[47]:

../../_images/notebooks_lattice_extensions_50_0.png

[48]:

lattice.extend([PrimeFactors(18)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[48]:

../../_images/notebooks_lattice_extensions_51_0.png

[49]:

lattice.extend([PrimeFactors(30)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[49]:

../../_images/notebooks_lattice_extensions_52_0.png

[50]:

lattice.extend([PrimeFactors(42)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[50]:

../../_images/notebooks_lattice_extensions_53_0.png

[51]:

lattice.extend([PrimeFactors(5040)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[51]:

../../_images/notebooks_lattice_extensions_54_0.png

[52]:

lattice.extend([PrimeFactors(7560)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[52]:

../../_images/notebooks_lattice_extensions_55_0.png

[53]:

lattice.extend([PrimeFactors(30240)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[53]:

../../_images/notebooks_lattice_extensions_56_0.png

[54]:

lattice.extend([PrimeFactors(60480)])
HasseDiagram[PrimeFactors](
    lattice,
    domain_renderer=PrimeFactorsRenderer(
        join_semi_lattice=True, meet_semi_lattice=True
    ),
)

[54]:

../../_images/notebooks_lattice_extensions_57_0.png