Notions of representability for cylindric algebras: some algebras are more representable than others
Authors: Sayed Ahmed Tarek 
Published: July 2022

The theory of cylindric algebras was introduced by Tarski in the fifties of the twentieth century, and its intensive study was further pursued by pioneers such as Henkin and Monk and, by the Hungarian mathematicians Andréka, Németi and Sain, and many of their students; to name only a few: Madarász, Marx, Kurucz, Simon, Mikulás, and Sági and many others outside Hungary including the author of this paper. Here we introduce and investigate new notions of representability for cylindric algebras and investigate various connections between such notions. Let $$2<n\le l<m\le \omega $$

2
<
n
≤
l
<
m
≤
ω

. Let $$\textsf {CA}_n$$

CA
n

denote the variety of cylindric algebras of dimension n and let $$\textsf {RCA}_n$$

RCA
n

denote the variety of representable $$\textsf {CA}_n$$

CA
n

s. We say that an atomic algebra $${{\mathfrak {A}}}\in \textsf {CA}_n$$

A
∈

CA
n

has the complex neat embedding property up to l and m if $${{\mathfrak {A}}}\in \textsf {RCA}_n\cap \textsf {Nr}_n\textsf {CA}_l$$

A
∈

RCA
n

∩

Nr
n

CA
l

and $${{{\mathfrak {C}}}{{\mathfrak {m}}}}\textsf {At}{{\mathfrak {A}}}\in \mathbf {S}\textsf {Nr}_n\textsf {CA}_m$$

C
m

At
A
∈
S

Nr
n

CA
m

. Fixing the prarameters l at the value n, this is a measure of how much the algebra is representable. The yardstick is how far can its Dedekind–MacNeille completion be dilated, that is to say, counting the number of more extra dimensions its Dedekind–MacNeille completion neatly embeds into. If $${{\mathfrak {A}}}, {{\mathfrak {B}}}\in \textsf {RCA}_n$$

A
,
B
∈

RCA
n

are atomic, $${{{\mathfrak {C}}}{{\mathfrak {m}}}}\textsf {At}{{\mathfrak {B}}}\in S\textsf {Nr}_n\textsf {CA}_l$$

C
m

At
B
∈
S

Nr
n

CA
l

and $${{{\mathfrak {C}}}{{\mathfrak {m}}}}\textsf {At}{{\mathfrak {A}}}\in S\textsf {Nr}_n\textsf {CA}_m$$

C
m

At
A
∈
S

Nr
n

CA
m

, then we say that $${{\mathfrak {A}}}$$
A
is more representable than $${{\mathfrak {B}}}$$
B
. When $$m=\omega $$

m
=
ω

, we say that $${{\mathfrak {A}}}$$
A
is strongly representable; this is the maximum degree of representability; the algebra in question cannot be ‘more representable’ than that. In this case the atom structure of $${{\mathfrak {A}}}$$
A
, namely $$\textsf {At}{{\mathfrak {A}}}$$

At
A

, is strongly representable in the sense of Hirsch and Hodkinson. This notion gives an infinite potential spectrum of ‘degrees’ of representability. In this connection, we exhibit various atomic algebras in $$\textsf {RCA}_n\cap \textsf {Nr}_n\textsf {CA}_l$$

RCA
n

∩

Nr
n

CA
l

that do no not have the complex neat embedding property for infinitely many values of l and m. It is known that the class of Kripke frames $$\textsf {Str}(\textsf {RCA}_n)=\{{{\mathfrak {F}}}: {{{\mathfrak {C}}}{{\mathfrak {m}}}}{{\mathfrak {F}}}\in \textsf {RCA}_n\}$$

Str

(

RCA
n

)

=

{
F
:

C
m

F
∈

RCA
n

}

is not elementary. From this it follows that there is some $$n<m<\omega $$

n
<
m
<
ω

such that $$\textsf {Str}(\mathbf {S}\textsf {Nr}_n\textsf {CA}_m)=\{{{\mathfrak {F}}}: {{{\mathfrak {C}}}{{\mathfrak {m}}}}{{\mathfrak {F}}}\in \mathbf {S}\textsf {Nr}_n\textsf {CA}_m\}$$

Str

(
S

Nr
n

CA
m

)

=

{
F
:

C
m

F
∈
S

Nr
n

CA
m

}

is not elementary. Replacing $$\mathbf {S}$$
S
by $$\mathbf {S}_c$$

S
c

(forming complete subalgebras), $$\mathbf {S}_d$$

S
d

(forming dense subalgebras) and $$\mathbf {I}$$
I
(forming isomorphic copies), respectively, we show that for any $$\mathbf {O}\in \{\mathbf {S}_c, \mathbf {S}_d, \mathbf {I}\}$$

O
∈
{

S
c

,

S
d

,
I
}

, the class of frames $$\textsf {Str}(\mathbf {O}\textsf {Nr}_n\textsf {CA}_{n+3})=\{{{\mathfrak {F}}}: {{{\mathfrak {C}}}{{\mathfrak {m}}}}{{\mathfrak {F}}}\in \mathbf {O}\textsf {Nr}_n\textsf {CA}_{n+3}\}$$

Str

(
O

Nr
n

CA

n
+
3

)

=

{
F
:

C
m

F
∈
O

Nr
n

CA

n
+
3

}

is not elementary. Metalogical applications are given to n-variable fragments of first-order logic endowed with so-called clique guarded semantics. The last semantics capture the new notions of representations introduced and studied in this paper.