Extension of semi-lattices and lattices

Extension of meet semi-lattices

[1]:

from galactic.algebras.lattice import MeetSemiLattice
from galactic.algebras.poset import HasseDiagram
from galactic.examples.arithmetic import IntegerRenderer
from galactic.examples.arithmetic import Integer

[2]:

meet_semi_lattice = MeetSemiLattice[Integer](
    domain=[
        Integer(3),
        Integer(7),
        Integer(2 * 7),
        Integer(3 * 5),
    ]
)
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[2]:

../../../_images/notebooks_algebras_lattice_extensions_2_0.png

[3]:

meet_semi_lattice.extend([Integer(3 * 7)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[3]:

../../../_images/notebooks_algebras_lattice_extensions_3_0.png

[4]:

meet_semi_lattice.extend([Integer(2 * 11)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[4]:

../../../_images/notebooks_algebras_lattice_extensions_4_0.png

[5]:

meet_semi_lattice.extend([Integer(11)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[5]:

../../../_images/notebooks_algebras_lattice_extensions_5_0.png

[6]:

meet_semi_lattice.extend([Integer(3 * 11)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[6]:

../../../_images/notebooks_algebras_lattice_extensions_6_0.png

[7]:

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

[7]:

../../../_images/notebooks_algebras_lattice_extensions_7_0.png

[8]:

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

[8]:

../../../_images/notebooks_algebras_lattice_extensions_8_0.png

[9]:

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

[9]:

../../../_images/notebooks_algebras_lattice_extensions_9_0.png

[10]:

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

[10]:

../../../_images/notebooks_algebras_lattice_extensions_10_0.png

[11]:

meet_semi_lattice.extend([Integer(5)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[11]:

../../../_images/notebooks_algebras_lattice_extensions_11_0.png

[12]:

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

[12]:

../../../_images/notebooks_algebras_lattice_extensions_12_0.png

[13]:

meet_semi_lattice.extend([Integer(2 * 5)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[13]:

../../../_images/notebooks_algebras_lattice_extensions_13_0.png

[14]:

meet_semi_lattice.extend([Integer(35)])
HasseDiagram[Integer](
    meet_semi_lattice, domain_renderer=IntegerRenderer(meet_semi_lattice=True)
)

[14]:

../../../_images/notebooks_algebras_lattice_extensions_14_0.png

[15]:

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

[15]:

../../../_images/notebooks_algebras_lattice_extensions_15_0.png

Extension of join semi-lattices

[16]:

from galactic.algebras.lattice import JoinSemiLattice
from galactic.algebras.poset import HasseDiagram
from galactic.examples.arithmetic import IntegerRenderer
from galactic.examples.arithmetic import Integer

[17]:

join_semi_lattice = JoinSemiLattice[Integer](
    domain=[
        Integer(3),
        Integer(7),
        Integer(2 * 7),
        Integer(3 * 5),
    ]
)
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[17]:

../../../_images/notebooks_algebras_lattice_extensions_18_0.png

[18]:

join_semi_lattice.extend([Integer(2 * 11)])
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[18]:

../../../_images/notebooks_algebras_lattice_extensions_19_0.png

[19]:

join_semi_lattice.extend([Integer(11)])
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[19]:

../../../_images/notebooks_algebras_lattice_extensions_20_0.png

[20]:

join_semi_lattice.extend([Integer(3 * 11)])
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[20]:

../../../_images/notebooks_algebras_lattice_extensions_21_0.png

[21]:

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

[21]:

../../../_images/notebooks_algebras_lattice_extensions_22_0.png

[22]:

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

[22]:

../../../_images/notebooks_algebras_lattice_extensions_23_0.png

[23]:

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

[23]:

../../../_images/notebooks_algebras_lattice_extensions_24_0.png

[24]:

join_semi_lattice.extend([Integer(5)])
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[24]:

../../../_images/notebooks_algebras_lattice_extensions_25_0.png

[25]:

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

[25]:

../../../_images/notebooks_algebras_lattice_extensions_26_0.png

[26]:

join_semi_lattice.extend([Integer(2 * 5)])
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[26]:

../../../_images/notebooks_algebras_lattice_extensions_27_0.png

[27]:

join_semi_lattice.extend([Integer(35)])
HasseDiagram[Integer](
    join_semi_lattice, domain_renderer=IntegerRenderer(join_semi_lattice=True)
)

[27]:

../../../_images/notebooks_algebras_lattice_extensions_28_0.png

[28]:

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

[28]:

../../../_images/notebooks_algebras_lattice_extensions_29_0.png

Extension of lattices

First example

[29]:

from galactic.algebras.lattice import Lattice
from galactic.algebras.poset import HasseDiagram
from galactic.examples.arithmetic import IntegerRenderer
from galactic.examples.arithmetic import Integer

[30]:

lattice = Lattice[Integer](
    domain=[
        Integer(3),
        Integer(7),
        Integer(2 * 7),
        Integer(3 * 5),
    ]
)
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[30]:

../../../_images/notebooks_algebras_lattice_extensions_32_0.png

[31]:

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

[31]:

../../../_images/notebooks_algebras_lattice_extensions_33_0.png

[32]:

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

[32]:

../../../_images/notebooks_algebras_lattice_extensions_34_0.png

[33]:

lattice.extend([Integer(11)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[33]:

../../../_images/notebooks_algebras_lattice_extensions_35_0.png

[34]:

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

[34]:

../../../_images/notebooks_algebras_lattice_extensions_36_0.png

[35]:

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

[35]:

../../../_images/notebooks_algebras_lattice_extensions_37_0.png

[36]:

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

[36]:

../../../_images/notebooks_algebras_lattice_extensions_38_0.png

[37]:

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

[37]:

../../../_images/notebooks_algebras_lattice_extensions_39_0.png

[38]:

lattice.extend([Integer(5)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[38]:

../../../_images/notebooks_algebras_lattice_extensions_40_0.png

[39]:

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

[39]:

../../../_images/notebooks_algebras_lattice_extensions_41_0.png

[40]:

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

[40]:

../../../_images/notebooks_algebras_lattice_extensions_42_0.png

[41]:

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

[41]:

../../../_images/notebooks_algebras_lattice_extensions_43_0.png

[42]:

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

[42]:

../../../_images/notebooks_algebras_lattice_extensions_44_0.png

[43]:

from galactic.algebras.lattice import ReducedContextDiagram

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

[43]:

../../../_images/notebooks_algebras_lattice_extensions_45_0.png

[44]:

from galactic.algebras.relational import BinaryTable

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

[44]:

	90090	60060	13860	15015	2730	4290	770	462
\(3=3^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)
\(7=7^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)
\(2=2^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(11=11^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(5=5^1\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(4620=2^23^15^17^111^1\)		\(\checkmark\)	\(\checkmark\)
\(195=3^15^113^1\)	\(\checkmark\)	\(\checkmark\)		\(\checkmark\)	\(\checkmark\)	\(\checkmark\)
\(6930=2^13^25^17^111^1\)	\(\checkmark\)		\(\checkmark\)

Second example

[45]:

lattice = Lattice[Integer](domain=[Integer(1)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[45]:

../../../_images/notebooks_algebras_lattice_extensions_48_0.png

[46]:

lattice.extend([Integer(2)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[46]:

../../../_images/notebooks_algebras_lattice_extensions_49_0.png

[47]:

lattice.extend([Integer(12)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[47]:

../../../_images/notebooks_algebras_lattice_extensions_50_0.png

[48]:

lattice.extend([Integer(18)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[48]:

../../../_images/notebooks_algebras_lattice_extensions_51_0.png

[49]:

lattice.extend([Integer(30)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[49]:

../../../_images/notebooks_algebras_lattice_extensions_52_0.png

[50]:

lattice.extend([Integer(42)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[50]:

../../../_images/notebooks_algebras_lattice_extensions_53_0.png

[51]:

lattice.extend([Integer(5040)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[51]:

../../../_images/notebooks_algebras_lattice_extensions_54_0.png

[52]:

lattice.extend([Integer(7560)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[52]:

../../../_images/notebooks_algebras_lattice_extensions_55_0.png

[53]:

lattice.extend([Integer(30240)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[53]:

../../../_images/notebooks_algebras_lattice_extensions_56_0.png

[54]:

lattice.extend([Integer(60480)])
HasseDiagram[Integer](
    lattice,
    domain_renderer=IntegerRenderer(join_semi_lattice=True, meet_semi_lattice=True),
)

[54]:

../../../_images/notebooks_algebras_lattice_extensions_57_0.png