International Journal of Computers, Communications & Control
Vol. I (2006), No. 1, pp. 81-99

Grigore C. Moisil (1906 - 1973) and his School in Algebraic Logic

George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

Abstract: We present in the paper a very concise but updated survey emphasizing the re-
search done by Gr. C. Moisil and his school in algebraic logic.
Keywords: n-valued Lukasiewicz-Moisil algebra, θ -valued Lukasiewicz-Moisil algebra,
Post algebra

The mathematical logic is one of the domain in which the creative spirit of Gr.C. Moisil manifested plenary.
His work in logic stands out by the the novelty, the variety and the depth of treated subjects. His first works are
connected to the top results of the time and wear an algebraic seal. The young professor from Jassy came after
a rich experience in mechanics and differential equations. Van der Waerden treatise of algebra has decisively
influenced his entry in logic by the algebraic gate. In the same time, these works have a powerful philosophical
imprint.

From this vast creation, the contributions in multiple-valued logics represent the part with the most intense
impact on today researches.

The first system of multiple-valued logic was introduced by J. Łukasiewicz in 1920. Independently, E. Post
introduced in 1921 a different multiple-valued logic. For Łukasiewicz, the motivation was of philosophical nature
- he was looking for an interpretation of the concepts of possibility and necessity - while for Post, the research was
intended as a natural mathematical generalization of bivalent logic.

In 1930, Łukasiewicz and Tarski studied a logic whose truth values are the real numbers from the interval [0, 1].

1 Łukasiewicz-Moisil algebras

In 1940, Gr. C. Moisil has defined the 3-valued and the 4-valued Łukasiewicz algebras and in 1942, the n-
valued Łukasiewicz algebras (n ≥ 2). His goal was to algebrize Łukasiewicz’s logic. Boolean algebras, algebraic
models of classical logic, are particular cases of that new structures.

In the description of a logical system, the implication was traditionally the principal connector. The n-valent
system of Łukasiewicz had as truth values the elements of the set

Ln =
{

0,
1

n−1 ,
2

n−1 , . . . ,
n−2
n−1 , 1

}

and was built around a new concept of implication, on which are based the definitions of the other connectors.
For Moisil, the basic structure is that of lattice, to which he adds a negation (getting the so called "De Morgan

algebra") and also some unary operations (called by Moisil "chryssipian endomorphisms"), representing the "nu-
ancing". The Łukasiewicz implication was let on a secondary plane and, in the case of an arbitrary valence, was
completely lost.

Further axiomatizations were suggested by A. Monteiro, R. Cignoli, C. Sicoe, S. Rudeanu and others.
An example of A. Rose from 1956 established that for n ≥ 5 the Łukasiewicz implication can no more be

defined on a Łukasiewicz algebra. Consequently, only for n = 3 and n = 4 the structures introduced by Moisil
are models for Łukasiewicz logic. The lost of implication has lead to another type of logic, called today "Moisil
logic", distinct from Łukasiewicz system; the logic corresponding to n-valued Łukasiewicz-Moisil algebras was
created by Moisil in 1964. The fundamental concept of Moisil logic is the nuancing.

Nowadays we feel it appropiate to call these algebras Łukasiewicz-Moisil algebras or LM algebras for short.
For complete information and references on Łukasiewicz-Moisil algebras see [25]
The work of Moisil on LM algebras covers two periods of time: a first period, during 1940-1942, when he intro-

duces the n-valued LM algebras with negation and studies special classes of these structures, as centered and axed
LM algebras and a second one, during 1954-1973, when he introduces the θ -valued LM algebras without negation,
applies multiple-valued logics to swiching theory and study algebraic properties of LM algebras (representation,
ideals, reziduation).

Copyright c© 2006 by CCC Publications



82 George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

Moisil’s works traced research directions for many Romanian and foreign mathematicians. In Argentina, at
Bahia Blanca, Antonio Monteiro and his school (Roberto Cignoli, Luiz Monteiro, Luiza Iturrioz, Maurice Abad
etc.) have contributed decisively to consolidate LM agebras as a domain of algebra of logic and to disseminate
them in the mathematical world.

In his PhD thesis from 1969 [29], R. Cignoli makes a very deep study of n-valued Moisil algebras (the name
he first gives to the n-valued Łukasiewicz algebras introduced by Moisil).

1.1 n-valued Łukasiewicz-Moisil algebras

The structure called "De Morgan algebra" was first studied by Moisil; the name was given by Antonio Monteiro
[142]; a duplicate name is "quasi-Boolean algebra" given by A. Bialynicki-Birula and H. Rasiowa.

Definition 1.1. A De Morgan algebra is a structure

(A,∨,∧,−, 0, 1)

such that (A,∨,∧, 0, 1) is a distributive lattice with 0 and 1 and the unary operation −, called negation, verifies:
(DM0) 1 = 0−,
(DM1) (x−)− = x,
(DM2) (x∧y)− = x− ∨y−.

Remark 1.2. In a De Morgan algebra we also have:
(DM3) (x∨y)− = x− ∧y−.

Definition 1.3. Let J = {1, 2, . . . , n−1}.
An n-valued Łukasiewicz-Moisil algebra (n ≥ 2) or an LMn bf algebra for short is an algebra

A = (A,∨,∧,−, (r j) j∈J , 0, 1)

of type (2, 2, 1, (1) j∈J , 0, 0) such that:

(i) (A,∨,∧,−, 0, 1) is a De Morgan algebra.
(ii) the unary operations r1, r2, . . . , rn−1 fulfil the following axioms: for every x, y ∈ A and every i, j ∈ J,

(L1) r j(x∨y) = r j x∨r j y,
(L2) r j x∨(r j x)− = 1,
(L3) r j ◦ri = ri,
(L4) r j(x−) = (rn− j x)−,

(L5) r1x ≤ r2x ≤ ··· ≤ rn−1x,
(L6) if r j x = r j y for every j ∈ J, then x = y; this is the determination principle.

If A fulfils (i) and only (L1)–(L5) we shall say that A is an LMn pre-algebra.

Proposition 1.4. In every LMn algebra A, the following properties are verified: for every x, y ∈ A and every j ∈ J,
(L7) r j(x∧y) = r j x∧r j y;
(L8) r j x∧(r j x)− = 0;
(L9) x ≤ y if and only if (r j x ≤ r j y, for every j ∈ J);
(L10) r1x ≤ x ≤ rn−1x;
(L11) r j 0 = 0, r j 1 = 1;
(L12) Let C(A) be the set of complemented elements of A, i.e.

C(A) = {x ∈ A | ∃x′ ∈ A, x∨x′ = 1, x∧x′ = 0}.

Let K j be the set of all elements of A left invariant by r j , j ∈ J, i.e.

K j = {x ∈ A | r j x = x}.



Grigore C. Moisil (1906 - 1973) and his School in Algebraic Logic 83

Then:
(i) r j x ∈ C(A), for every j ∈ J, x ∈ A and
(ii) C(A) = K j , for every j ∈ J;

(L12’) (C(A),∨,∧,−, 0, 1) is a Boolean algebra, where x− = x′;
(L12”) If z ∈ C(A), then for every x ∈ A:

x∧z = 0 ⇐⇒ x ≤ z−,
z∨x = 1 ⇐⇒ z− ≤ x;

(L13) x− ∨rn−1x = 1;
(L14) x∧(rn−1x)− = 0.

Example 1.5. The algebra

Ln = L
(LMn)
n = (Ln,∨,∧,−, (r j) j∈J , 0, 1),

where

Ln =
{

0,
1

n−1 ,
2

n−1 , . . . ,
n−2
n−1 , 1

}

and 



x∨y = max(x, y), x∧y = min(x, y), x− = 1−x,
r j

(
i

n−1
)

=
{

0, if j + i < n,
1, if j + i ≥ n, i ∈ {0}∪J, j ∈ J,

is an LMn algebra, that we shall call the canonical LMn algebra.
The proper subalgebras of Ln have the form:

S = Ln −
⋃

x∈Ln−{0}
{x, x−}.

They are LMn algebras.
The smallest subalgebra of Ln (with respect to ⊆) is C(Ln) = {0, 1}, which is also a Boolean algebra, cf.

(L12’).
For instance, the subalgebras of

- L3 are L2 and L3,
- L4 are L2 and L4 and
- L5 are L2, L3, {0, 1/4, 3/4, 1} and L5.
Remark 1.6. LM2 algebras coincide with Boolean algebras.

Proposition 1.7. In every LMn pre-algebra, the determination principle (L6) is equivalent to each of the following
conditions: for every x, y ∈ L,

(a) x∧(r j x)− ∧r j+1y ≤ y, for every j ∈ J −{n−1};

(b) x∧
n−1∧
j=1

((r j x)− ∨r j y) ≤ y.

þ (Representation theorem of Moisil)
Every LMn algebra can be embedded in a direct product of copies of the canonical LMn algebra Ln.

Corollary 1.1. Every LMn algebra is a subdirect product of subalgebras of the canonical LMn algebra Ln.

∗∗∗
In 1968, Gr. C.Moisil introduced the θ -valued Łukasiewicz algebras or LMθ algebras for short (without

negation), where θ is the order type of a chain with first and last element. The concept of θ -valued Łukasiewicz
algebra is obtained from that of n-valued, on the one hand, by dropping the negation − and on the other hand, by
replacing the set Ln by a totally ordered set I with first and least elements and by adapting the axioms to this case;
the Determination Principle is preserved. These structures were thought by Moisil as models of a logic with an
infinity of nuances. According to a confession done by Moisil, he imagined LMθ algebras (without negation) long



84 George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

time ago, but the care of finding a strong motivation for them delayed the announcement; the motivation was found
when Moisil met Zadeh’s fuzzy set theory, in which he saw a confirmation of his old ideas.

In 1969, Marek and Traczyk [110] introduced the notion of generalized Łukasewicz algebra (with negation),
in an attempt to generalize to the infinite case the LMn algebras; but their generalization is not a natural one.

In his PhD thesis from 1972 [64], G. Georgescu studied duality theory for Moisil’s LMθ algebras (without
negation), the injective objects (and their characterization), monadic and poliadic algebras.

In his PhD thesis from 1981 [53], A. Filipoiu studied the LMθ algebras (without negation) and their associated
logic. He gives a representation theorem for LMθ algebras by aids of θ -valent Moisil field.

In his Master thesis from 1981 [12] also, L. Beznea studies a generalization of LMθ algebras (without negation)
obtained by eliminating the Determination Principle.

Later on, in his PhD theses from 1984 [21], V. Boicescu introduced and studied the n-valued LM algebras
without negation, as a particular case of Moisil’s LMθ algebras (without negation).

Following the inverse way, A. Iorgulescu, in her PhD thesis from 1984 [90] also, introduced and studied a
natural generalization of Moisil’s LMn algebras to the infinite case, called θ -valued LM algebras with negation
or LMθ algebras with negation for short; any LMθ algebra with negation is a Moisil’s LMθ algebra without
negation.

2 Connection with logic

Gr. C. Moisil invented LM algebras in order to create an algebraic structure playing the same role with respect
to the multiple-valued logic as Boolean algebras play with respect to classical, bivalent logic. However, as shown
by the example of A. Rose, this only happens for the cases n = 3 and n = 4.

The algebraic structures adequate to the infinite-valued logic of Łukasiewicz (truth valued in the real interval
[0, 1]) are the MV-algebras introduced by C.C. Chang in 1958 or, equivalently, the Wajsberg algebras introduced by
Font, Rodriguez and Torrens in 1984; D. Mundici proved in in 1986 that MV algebras are categorically equivalent
to lattice-ordered Abelian groups with strong unit.

R. Grigolia’s MVn algebras, introduced in 1977, and Cignoli’s proper Łukasiewicz algebras, introduced in
1982, are algebraic structures corresponding to n-valued logic of Łukasiewicz.

The logic corresponding to LMn algebras was created by Moisil himself in 1964. Łukasiewicz logic has im-
plication as its primary connector, while Moisil logic is based on the idea of nuance, expressed algebraically by
the Chrysippian endomorphisms. The “engine" of the latter logic is Moisil’s Determination Principle, according to
which an n-valued sentence is determined by its Boolean nuances. The Determination Principle realizes a transfer
from the multiple-valued logic to the classical logic. This determination brings Moisil logic much closer to clas-
sical logic than Łukasiewicz logic. One could say that Moisil logic is derived from classical logic by the idea of
nuancing. Algebraically, this tight relationship is expressed by the fundamental adjunction between the categories
of Boolean and Łukasiewicz algebras.

V. Boicescu in 1971 and A. Filipoiu in 1981 introduced and studied logics appropiate to LMθ algebras without
negation. (i.e. infinite-valued LM algebras).

A. Filipoiu generalized Smullyan’s method of analytic tableaux to θ -valued logic without negation and studied
the θ -valued predicate calculus as well, with applications to systems of recording and retrieval of information.

Łukasiewicz logic, Post logic and Moisil logic consitute the three directions in the classical theory of multiple-
valued logic. Their corresponding algebraic models are MV algebras, Post algebras and LM algebras.

3 Connections with other structures of algebraic logic

Moisil introduced in 1941 the centered LM3 algebras.
Post algebras (cf. P. Rosenbloom (1942), G. Epstein (1960), T. Traczyk (1963)) turn out to be centered LM

algebras, cf. R. Cignoli (1969) and G. Georgescu - C. Vraciu (1969). The θ -valued Post algebras were studied by
T. Traczyk (1967) and G. Georgescu (1971).

Gr. C. Moisil, R. Cignoli, L. Iturrioz, A. Monteiro and V. Boicescu studied LM algebras as particular cases of
Heyting algebras. V. Boicescu also studied LM algebras as Stone algebras.



Grigore C. Moisil (1906 - 1973) and his School in Algebraic Logic 85

LM3 algebras and LM4 algebras are polynomially equivalent to MV3 algebras and MV4 algebras, respectively,
since they are the algebraic counterpart of the 3-valued Łukasiewicz logic and the 4-valued Łukasiewicz logic,
respectively. D. Mundici was first to point out the equivalence between LM3 algebras and MV3 algebras, in 1989.
Then A. Iorgulescu, in 1998-2000 [91] - [94], pointed out the isomorphism between the categories of LMk algebras
and of MVk algebras, for k = 3, 4 and also studied the categories LMn and MVn for n ≥ 5, showing that every MVn
can be made into an LMn algebra. She then studied those LMn algebras that can be viewed as MVn algebras:

3.1 Connections between LMn algebras and MVn algebras

MV algebras were introduced by C.C. Chang, in 1958 [26]. A simplified list of axioms of MV algebras was
given by Mangani [109], as follows:

Definition 3.1. An MV algebra is an algebra

A = (A,⊕,−, 0)

of type (2, 1, 0), where the following axioms are verified: for every x, y, z ∈ A,
(MV1) (A,⊕, 0) is an Abelian monoid,
(MV2) x⊕0− = 0−,
(MV3) (x−)− = x,
(MV4) (x− ⊕y)− ⊕y = (y− ⊕x)− ⊕x,
where x ·y = (x− ⊕y−)−.

Definition 3.2. For any m ∈ IN, we have:
(i) 0x = 0 and (m + 1)x = mx⊕x,
(ii) x0 = 1 and xm+1 = xm ·x.

The MVn algebras were introduced by Revaz Grigolia in 1977 [87], as follows.

Definition 3.3. An MVn algebra (n ≥ 2) is an MV algebra A = (A,⊕,−, 0), whose operations fulfil the additional
axioms:

(M1) (n−1)x⊕x = (n−1)x,
(M1’) xn−1 ·x = xn−1

and, if n ≥ 4, the axioms:
(M2) [( jx)·(x− ⊕[( j −1)x]−)]n−1 = 0,
(M2’) (n−1)[x j ⊕(x− ·[x j−1]−)] = 1,

where 1 < j < n−1 and j does not divide n−1.
Corollary 3.1. MV2 algebras coincide with Boolean algebras.

Example 3.4. The MV algebra Ln = L
(MVn)
n = (Ln,⊕,−, 0), where

Ln =
{

0,
1

n−1 ,
2

n−1 , . . . ,
n−2
n−1 , 1

}

and for any x, y ∈ Ln:

x⊕y = min(1, x + y), x ·y = max(0, x + y−1), x− = 1−x

and
x∨y = max(x, y), x∧y = min(x, y),

is an MVn algebra. We shall call it the canonical MVn algebra.
Note that B(Ln) = {0, 1}.
The subalgebras of Ln are of the form:

Sm =
{

0,
K

n−1 , . . . ,
(m−2)K

n−1 , 1
}

,



86 George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

where K = n−1m−1 , if m−1 divides n−1.
The subalgebras Sm of Ln are isomorphic to Lm =

{
0, 1m−1 , . . . ,

m−2
m−1 , 1

}
, if m − 1 divides n − 1, and they are

MVn algebras.
Hence Lm = (Lm,⊕,·,−, 0, 1) (m ≤ n) is an MVn algebra if and only if m−1 divides n−1.
For instance, the subalgebras of:

- L3 are L2 and L3,
- L4 are L2 and L4 and
- L5 are L2, L3 and L5.

þ
Every MVn algebra is a subdirect product of subalgebras of the canonical MVn algebra Ln.
D. Mundici was the first to prove in 1989 [152] that MV3 algebras coincide with LM3 algebras.
A. Iorgulescu has proved in 1998-2000 [91] - [94] that:

1 - MV4 algebras coincide with LM4 algebras,
2 - the canonical MVn algebra coincides with the canonical LMn algebra (n ≥ 2),
3 - for n ≥ 5, any MVn algebra is a LMn algebra,
4 - those LMn algebras which are MVn algebras, for every n ≥ 5, are exactly Cignoli’s proper n-valued Łukasiewicz
algebras.

Here are for short the results 1-3:
∗∗∗

To obtain the transformation of an MVn algebra into an LMn algebra, for any n ≥ 3, Iorgulescu used Suchoń’s
transformation [174]:

Suchoń defines Moisil operators (σ j) j∈J ( σ j = rn− j ) of the canonical LMn algebra (n ≥ 3) starting from the
Łukasiewiczian implication → and from the negation −. He puts

B3(x) = (x
−) → x and B j+1(x) = (x−) → B j(x), j ≥ 3. (1)

Then he defines:
σ1x = Bn(x) (2)

and for 1 < j ≤ [n/2], σ j x =
{

σn−1(Bl+1(x)), l j ≥ n−1
σl j(Bl+1(x)), l j < n−1, (3)

where l = max{m | m( j −1) < n−1},

while σn− j(x) = (σ j(x−))−, for 1 ≤ j ≤ [n/2]. (4)
Suchoń’s Moisil operators verify: σ1 ≥ σ2 ≥ . . . ≥ σn−1.
Remark 3.5. If we want to use Suchoń’s construction, it is convenient to consider not the MV algebra (A,⊕,−, 0),
but the Wajsberg algebra, (A,→,−, 1), introduced by J. M. Font, A. J. Rodriguez and A. Torrens in 1984; MV
algebras and Wajsberg algebras are isomorphic structures:

• if A = (A,→,−, 1) is a Wajsberg algebra and if we define
α(A) = (A,⊕,−, 0)

by
x⊕y = x− → y, 0 = 1−, (5)

then α(A) is an MV algebra.
• Conversely, if A = (A,⊕,−, 0) is an MV algebra and if we define

β (A) = (A,→,−, 1)
by

x → y = x− ⊕y, (6)
1 = 0−,

then β (A) is a Wajsberg algebra.
• The maps α, β are mutually inverse.



Grigore C. Moisil (1906 - 1973) and his School in Algebraic Logic 87

It follows immediately by (1) that

B3(x) = x⊕x = 2x and B j+1(x) = x⊕B j(x) = jx, j ≥ 3. (7)
By using Suchoń’s construction, Iorgulescu then gave the following

Definition 3.6. Let A = (A,⊕,−, 0) be an MVn algebra (n ≥ 3). Define
ΦS(A) = (A,∨,∧,−, (r j) j∈J , 0, 1)

by
x∨y = x ·y− ⊕y, x∧y = (x− ∨y−)−,

rn−1x = (n−1)x, (8)
rn− j x =

{
r1(lx), l j ≥ n−1

rn−l j(lx), l j < n−1, (9)
for 1 < j ≤ [n/2], l = max{m | m( j −1) < n−1} ,

r j x = (rn− j(x−))−, 1 ≤ j ≤ [n/2]. (10)
Proposition 3.7. If Ln is the canonical MVn algebra (n ≥ 3), then ΦS(Ln) is the canonical LMn algebra.

þ If A is an MVn algebra (n ≥ 3), then ΦS(A) is an LMn algebra.
∗∗∗

Proposition 3.8. 1) Given the canonical LMn algebra (n ≥ 3)
Ln = (Ln,∨,∧,−, (r j) j∈J , 0, 1),

define Ψ(Ln) = (Ln, ⊕n ,−, 0) by :
if n = 2k + 1,

x⊕2k+1 y = (x∨r2ky)∧(y∨r2kx) (11)
∧ (x∗ ∨r2k−1y)∧(y∗ ∨r2k−1x)

...

∧ (x(k−1)∗ ∨rk+1y)∧(y(k−1)∗ ∨rk+1x),
if n = 2k,

x⊕2k y = (x∨r2k−1y)∧(y∨r2k−1x) (12)
∧ (x∗ ∨r2k−2y)∧(y∗ ∨r2k−2x)

...

∧ (x(k−1)∗ ∨rky)∧(y(k−1)∗ ∨rkx),
where x∗ is the successor of x and

x2∗ = (x∗ )∗ , xm∗ =
(

x(m−1)∗
)∗

(13)

Then Ψ(Ln) is the canonical MVn algebra.

2) The maps ΦS, from Proposition 3.7, and Ψ are mutually inverse.
Since for n = 3, in the canonical LM3 algebra L3, the operation ⊕ is:

x⊕y = (x∨r2y)∧(y∨r2x)
and for n = 4, in the canonical LM4 algebra L4, the operation ⊕ is defined by:

x⊕y = (x∨r3y)∧(y∨r3x)∧(x∗ ∨r2y)∧(y∗ ∨r2x) =
= (x∨r3y)∧(y∨r3x)∧(x− ∨y− ∨r2x∨r2y),

it follows that the transformation Ψ is not polynomial for n ≥ 5).
Those LMn algebras which are MVn algebras (i.e. for which the transformation Ψ is defined), for every n ≥ 5,

are exactly Cignoli’s proper n-valued Łukasiewicz algebras [34], but the proof is very technical [94].



88 George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

4 Representation theorems

Numerous representation theorems have been given for LM algebras. The first is due to Moisil himself and
is reminiscent of the representation theorem for Boolean algebras: every LMn algebra can be embadded into a
Cartesian power of Ln.

In a modern vision [25], every LM algebra is a subdirect product of subalgebras of the algebra In(I, L2) of
incresing functions from I to L2. In particular every LMn algebra is a subdirect product of subalgebras of Ln
(Cignoli). L is a direct product of subalgebras of Ln if and only if it is complete and atomic (Boicescu 1984).

The representation by continuous functions studied by Cignoli, Boicescu and Filipoiu, means that for every
LMθ algebra without negation L there is a unique Boolean space X such that L is isomorphic to the algebra of all
continuous functions f : X → In(I, L2), where In(I, L2) is endowed with the topology having as basis the principal
ideals and the principal filters generated by the characteristic functions of the sets {k | k > α}, α ∈ θ .

The representation of LMθ algebras without or with negation by Moisil fields of sets is due to Filipoiu. The
Stone duality was extended from Boolean algebras to LMθ algebras without or with negation with the aid of a
suitable concept called LMθ -valued Stone space (Cignoli, Georgescu, Iorgulescu), while the Priestley duality is
based on a suitable adaptation of the concept of Priestley space (Filipoiu).

The representation of LM algebras as algebras of fuzzy sets was studied by D. Ponasse, J.L. Coulon and J.
Coulon, S. Ribeyre and S. Rudeanu.

The representation of LMn algebras by LM3 algebras is legitimated by the "good" properties of the latter and
was studied by A. Monteiro, L. Monteiro, F. Coppola, V. Boicescu and A. Iorgulescu.

5 Categorial aspects

The Stone and Priestley dualities are in fact equivalences of categories. Other categorial properties of LM
algebras were studied. Here are a few samples.

The association of L with the Boolean alegebra C(L) of complemented elements of L is extended to a functor
C :LMθ →B, while the association of a Boolean algebra B with the algebra In(I, B) is extended to a functor T :
B →LMθ . Then C and T are adjoint functors, C is faithful and T is fully faithful. This yields in particular the
representation theorem of Moisil. The construction of the functors C and T was given by Moisil himself.

The injective and projective objects have also been studied, for instance, an LMθ algebra is injective if and
only if it is a complete Post algebra (whose center is a complete Boolean algebra), cf. L. Monteiro, R. Cignoli, G.
Georgescu and C. Vraciu, V. Boicescu.

6 Ideals and congruences

The study of the appropiate ideal and congruence theory for LM algebras was undertaken by Gr. C. Moisil,
A. Monteiro, R. Cignoli, C. Sicoe. V. Boicescu introduced the concepts of θ -ideal and θ -congruence, the prime
spectre. For instance, in the case of LMn algebras without negation, the congruence lattice of L is a Boolean
algebra (a Stone algebra) if and only if L is finite (C(L) is a complete Boolean algebra).

7 Monadic and polyadic algebras

L. Monteiro and G. Georgescu studied the generalization to LM algebras of the monadic and polyadic Boolean
algebras introduced by P.R. Halmos. Sample results: the representation of monadic LM algebras by functional
monadic LM algebras and the semantic completeness for polyadic LM algebras. A paper of G. Geogescu, A.
Iorgulescu and I. Leuştean investigates monadic MVn algebras and closed MVn algebras.



Grigore C. Moisil (1906 - 1973) and his School in Algebraic Logic 89

8 Miscellanea

Various other topics have also been studied. Thus:
V. Boicescu proved that the lattice of equational subclass of LMn is a finite Heyting algebra. The study of

atomic algebras and the characterization of simple algebras as subalgebras of In(I, L2) and the property that LMn
algebras without negation form an equational class, are also due to Boicescu. The study of irredundant algebras
and of exactly n-valued algebras is due to Boicescu as well.

A. Iorgulescu introduced and studied m-complete LMθ algebras with negations, generalizing many of the prop-
erties of m-complete Boolean algebras.

G. Georgescu and I. Leuştean studied probabilities on LM algebras.
L. Beznea studied a generalization of LM algebras, obtained by dropping the determination principle.
Let us also mention M. Sularia’s theory of D algebras. These structures are subdirect products between a

Heyting algebra and a Brouwer algebra and represent the algebraic counterpart of a logic of problem solving.
In [44], [48], the authors study Łukasiewicz BCK algebras endowed with Moisil oparators.
In [163], [164], C. Sanza introduced and studied (monadic) n×m-valued Łukasiewicz algebras with negation.
In a very recent paper [103], I. Leuştean proposes a unifying framework for LMn algebras, MV algebras and

Post algebras; essentially, an LMn+1 algebra is charcterized by a string of n Boolean ideals of his Boolean center.
The necessary and sufficient conditions are given that such a string must satisfy to define a MVn+1 algebra or a
Post algebra of order n + 1. This result could be seen as a generalization of Moisil’s Determination Principle. As
an application, in paper [75], some special Cauchy completions of MVn+1 algebras are characterized by using the
properties of corresponding strings of Boolean ideals.

In another very recent paper [75], G.Georgescu and A. Popescu introduced the notion of n-nuanced MV
algebra, by performing a Łukasiewicz-Moisil nuancing construction on top of MV-algebras. These structures
extend both MV-algebras and Łukasiewicz-Moisil algebras, thus unifying two important types of structures in
the algebra of logic. On a logical level, n-nuanced MV algebras amalgamate two distinct approaches to many-
valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that
of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than truth degrees. They study
n-nuanced MV algebras mainly from the algebraic and categorial points of view and also consider some basic
model-theoretic aspects. The relationship with a suitable notion of n-nuanced ordered group via an extension of
the Γ construction is also analyzed:

8.1 n-nuanced MV algebras

Usually, MV algebras are defined only in terms of ⊕, − and 0. However, in order to point out the symmetry of
these structures, the authors prefered the following slightly redundant definition:

Definition 8.1. An MV algebra is a structure (A,⊕,¯,−, 0, 1), satisfying the following axioms:
(MV1’) (A,⊕, 0) and (A,¯, 1) are commutative monoids,
(MV2’) x¯0 = 0 and x⊕1 = 1,
(MV3’) (x−)− = x,
(MV4’) (x⊕y)− = x− ¯y−,
(MV5’) (x¯y−)⊕y = (y¯x−)⊕x.

Definition 8.2. A generalized De Morgan algebra is a structure L = (L,⊕,¯,−, 0, 1), where ⊕, ¯ are binary
operations, − is a unary operation, and 0, 1 are constants such that the following conditions hold:
(i) (L,⊕, 0), (L,¯, 1) are commutative monoids;
(ii) (x⊕y)− = x− ¯y− and (x−)− = x for all x, y ∈ L; .
Remark 8.3. If L is a generalized De Morgan algebra, then (x¯y)− = x− ⊕y− for all x, y ∈ L.
Definition 8.4. An n-nuanced MV-algebra (NMVn algebra for short) is a structure

L = (L,⊕,¯,−, r1, . . . , rn−1, 0, 1)

such that (L,⊕,¯,−, 0, 1) is a generalized De Morgan algebra and r1, . . . , rn−1 satisfy the following axioms:
(A0) rix⊕((rix)− ¯riy) = riy⊕((riy)− ¯rix), for i ∈ {1, . . . , n−1},



90 George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

(A1) ri(x⊕y) = rix⊕riy, ri(x¯y) = rix¯riy, ri(0) = 0, ri(1) = 1, for i ∈ {1, . . . , n−1},
(A2) rix⊕(rix)− = 1, rix¯(rix)− = 0, for i ∈ {1, . . . , n−1},
(A3) ri ◦r j = r j , for i, j ∈ {1, . . . , n−1},
(A4) ri(x−) = (rn−ix)−, for i ∈ {1, . . . , n−1},
(A5) (Determination Principle:) if rix = riy for each i ∈ {1, . . . , n−1}, then x = y,
(A6) r1x ≤ r2x ≤ . . . ≤ rn−1x.

Remark 8.5. NMVn algebras provide a common generalization of MV- and Łukasiwicz-Moisil algebras. Indeed,
- if n = 2, then, because of the Determination Principle, r1 is the identity, thus an NMVn algebra can be identified
with an MV-algebra;
- if (L,⊕,¯,−, 0, 1) is a De Morgan algebra, then (L,⊕,¯,−, r1, . . . , rn−1, 0, 1) becomes an LMn algebra.
Example 8.6. Let A = (A,⊕,¯,−, 0, 1) be an MV-algebra. Consider the set

T (A) = {(x1, . . . , xn−1) ∈ An−1 | x1 ≤ . . . ≤ xn−1}.

Since An−1 is an MV-algebra (with operations taken component-wise from A) and T (A) is closed under the
operations 0, 1, ⊕, ¯ (where 0 and 1 are the constant vectors), then we can consider these operations on T (A).
We furthermore define −, r1, . . . , rn−1 by:

(x1, . . . , xn−1)− = (x−n−1, . . . , x
−
1 ),

ri(x1, . . . , xn−1) = (xi, . . . , xi), for i ∈ {1, . . . , n−1}.
Then (T (A),⊕,¯,−, r1, . . . , rn−1, 0, 1) is an NMVn algebra.

Define
M(L) = {x ∈ L | rix = x for all i ∈ {1, . . . , n−1}}.

Then M(L), together with the operations ⊕,¯,−, 0, 1 induced by L, is an MV algebra, called the MV-center
of L.

In the MV-algebra M(L) we have a canonical order ≤. Let us define an extension of this order to L by:

x ≤ y iff, for each i ∈ {1, . . . , n−1}, rix ≤ riy.

Because of the Determination Principle, this is indeed an order and because of (A3), it is indeed an extension
of the order on M(L). Moreover, the compatibility properties listed in the following lemma are obvious:

Proposition 8.7. The following properties are true in a L:
(1) 0 is the greatest and 1 is the least element in L w.r.t. ≤;
(2) for each x, y ∈ L, x ≤ y iff y− ≤ x−;
(3) for each x, x′, y, y′ ∈ L, if x ≤ x′ and y ≤ y′, then x⊕y ≤ x′ ⊕y′ and x¯y ≤ x′ ¯y′,
(4) r1x ≤ x ≤ rn−1x, for any x ∈ L,
(5) for x, y ∈ L, if x⊕y = 1 and x¯y = 0, then x, y ∈ M(L) and y = x−,
(6) M(L) = {x ∈ L | x⊕x− = 1′, x¯x− = 0}.

9 Applications to switching theory

Whereas Boolean algebra is a suitable tool for the study of networks made up of binary devices, the study of
networks involving multi-positional devices and the so-called hazard and race phenomena have imposed the use of
other algebraic tools, namely Galois fields, Łukasiewicz-Moisil algebras and the theory of discrete functions.

Moisil investigated circuits involving devices such as polarized relays with unstable neutral, ordinary relays
under low self-maintaining current, valves, resistances, multi-positional relays, as well as transistors and other
electronic devices. See also [125]. Moisil has described the operation of such devices by characteristic equations
of the form xk+1 = ϕ(ξk, xk), where the variable x associated with the relay contact takes values in Ln, where n ≤ 5



Grigore C. Moisil (1906 - 1973) and his School in Algebraic Logic 91

depends on the type of the relay, ξ ∈ L2 is a variable associated with the current and the index k or k + 1 indicates
the value of the corresponing variable at time t = k or t = k + 1, respectively.

The synthesis problem consists in designing a circuit made up of several relays and whose operation be
described by a given equation of the form

Xk+1 = F(Ak, Xk), (14)

where X is the vector of the variables x associated with the relays of the circuit, A is the input vector and the
meaning of the index k or k + 1 is the same as above. To solve this problem, Moisil notices the crucial point that
the structure of such a circuit is determined by the expression of a function G which satisfies the identity

Ξ = G(A, X ), (15)

where Ξ is the vector of the variables ξ associated with the relay of the circuit. So if

Xk+1 = Phi(Ξk, Xk) (16)

is the vector form of the characteristic equations of the relays of the circuit, it follows from (14) and (15) that

F(Ak, Xk) = Φ(Ξk, Xk), (17)

for any k. Therefore (15) transforms (17) into the identity

F(A, X ) = Φ(G(A, X ), X ), (18)

and Moisil’s method for solving the synthesis problem consists in solving the functional equation (18) with respect
to G.

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98 George Georgescu, Afrodita Iorgulescu, Sergiu Rudeanu

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George Georgescu
University of Bucharest

Address: Str. Academiei No. 14, Romania
E-mail: georgescu@funinf.cs.unibuc.ro

Afrodita Iorgulescu
Academy of Economic Studies

Address: Piaţa Romană Nr. 6 - R 70167,
Oficiul Poştal 22, Bucharest, Romania

E-mail: afrodita@ase.ro

Sergiu Rudeanu
University of Bucharest

Address: Str. Academiei No. 14, Romania
E-mail: rud@funinf.cs.unibuc.ro