<s>
A	O
subsumption	B-Application
lattice	I-Application
is	O
a	O
mathematical	O
structure	O
used	O
in	O
the	O
theoretical	O
background	O
of	O
automated	B-Application
theorem	I-Application
proving	I-Application
and	O
other	O
symbolic	B-Algorithm
computation	I-Algorithm
applications	O
.	O
</s>
<s>
This	O
lattice	O
is	O
called	O
the	O
subsumption	B-Application
lattice	I-Application
.	O
</s>
<s>
The	O
join	O
and	O
the	O
meet	O
operation	O
in	O
this	O
lattice	O
are	O
called	O
anti-unification	B-Application
and	O
unification	B-Algorithm
,	O
respectively	O
.	O
</s>
<s>
If	O
f	O
is	O
a	O
binary	O
function	O
symbol	O
,	O
g	O
is	O
a	O
unary	O
function	O
symbol	O
,	O
and	O
x	O
and	O
y	O
denote	O
variables	O
,	O
then	O
the	O
terms	O
f(x,y )	O
,	O
f(g(x )	O
,	O
y	O
)	O
,	O
f(g(x )	O
,	O
g(y )	O
)	O
,	O
f(x,x )	O
,	O
and	O
f(g(x )	O
,	O
g(x )	O
)	O
form	O
the	O
minimal	O
non-modular	O
lattice	O
N5	O
(	O
see	O
Pic.1	O
)	O
;	O
its	O
appearance	O
prevents	O
the	O
subsumption	B-Application
lattice	I-Application
from	O
being	O
modular	O
and	O
hence	O
also	O
from	O
being	O
distributive	O
.	O
</s>
<s>
The	O
set	O
of	O
linear	O
terms	O
,	O
that	O
is	O
of	O
terms	O
without	O
multiple	O
occurrences	O
of	O
a	O
variable	O
,	O
is	O
a	O
sub-poset	O
of	O
the	O
subsumption	B-Application
lattice	I-Application
,	O
and	O
is	O
itself	O
a	O
lattice	O
.	O
</s>
<s>
their	O
anti-unification	B-Application
and	O
unification	B-Algorithm
,	O
corresponds	O
to	O
intersection	O
and	O
union	O
of	O
their	O
path	O
sets	O
,	O
respectively	O
.	O
</s>
<s>
Apparently	O
,	O
the	O
subsumption	B-Application
lattice	I-Application
was	O
first	O
investigated	O
by	O
Gordon	O
D	O
.	O
Plotkin	O
,	O
in	O
1970	O
.	O
</s>
