The Finite Embeddability Property for IP Loops and Local - - PowerPoint PPT Presentation

The Finite Embeddability Property for IP Loops and Local Embeddability of Groups into Finite IP Loops

Martin Vodiˇ cka Max-Planck Institute, Leipzig Pavol Zlatoˇ s Comenius University, Bratislava LOOPS 2019 Budapest July 8 – 13, 2019

The Finite Embeddability Property (FEP), was introduced by Henkin [1956].

The Finite Embeddability Property (FEP), was introduced by Henkin [1956]. A class K of groupoids has the FEP if for every (G, ·) ∈ K and each finite set X ⊆ G there is a finite (H, ∗) ∈ K such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X.

The Finite Embeddability Property (FEP), was introduced by Henkin [1956]. A class K of groupoids has the FEP if for every (G, ·) ∈ K and each finite set X ⊆ G there is a finite (H, ∗) ∈ K such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. A more general notion of local embeddability can be traced back to a paper by Mal ’tsev [1941].

The Finite Embeddability Property (FEP), was introduced by Henkin [1956]. A class K of groupoids has the FEP if for every (G, ·) ∈ K and each finite set X ⊆ G there is a finite (H, ∗) ∈ K such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. A more general notion of local embeddability can be traced back to a paper by Mal ’tsev [1941]. It was reintroduced and studied in detail mainly for groups by Gordon and Vershik [1998]:

The Finite Embeddability Property (FEP), was introduced by Henkin [1956]. A class K of groupoids has the FEP if for every (G, ·) ∈ K and each finite set X ⊆ G there is a finite (H, ∗) ∈ K such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. A more general notion of local embeddability can be traced back to a paper by Mal ’tsev [1941]. It was reintroduced and studied in detail mainly for groups by Gordon and Vershik [1998]: A groupoid (G, ·) is locally embeddable into a class of groupoids M if for every finite set X ⊆ G there is (H, ∗) ∈ M such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X.

The Finite Embeddability Property (FEP), was introduced by Henkin [1956]. A class K of groupoids has the FEP if for every (G, ·) ∈ K and each finite set X ⊆ G there is a finite (H, ∗) ∈ K such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. A more general notion of local embeddability can be traced back to a paper by Mal ’tsev [1941]. It was reintroduced and studied in detail mainly for groups by Gordon and Vershik [1998]: A groupoid (G, ·) is locally embeddable into a class of groupoids M if for every finite set X ⊆ G there is (H, ∗) ∈ M such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. Informally, every finite cut-out from the multiplication table of (G, ·) can be embedded into some grupoid from M.

The Finite Embeddability Property (FEP), was introduced by Henkin [1956]. A class K of groupoids has the FEP if for every (G, ·) ∈ K and each finite set X ⊆ G there is a finite (H, ∗) ∈ K such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. A more general notion of local embeddability can be traced back to a paper by Mal ’tsev [1941]. It was reintroduced and studied in detail mainly for groups by Gordon and Vershik [1998]: A groupoid (G, ·) is locally embeddable into a class of groupoids M if for every finite set X ⊆ G there is (H, ∗) ∈ M such that X ⊆ H and x · y = x ∗ y for all x, y ∈ X satisfying x · y ∈ X. Informally, every finite cut-out from the multiplication table of (G, ·) can be embedded into some grupoid from M. Equivalently, (G, ·) can be embedded into an ultraproduct of grupoids from M.

A class K has the FEP iff if every (G, ·) ∈ K is locally embeddable into the class Kfin of all finite members in K.

A class K has the FEP iff if every (G, ·) ∈ K is locally embeddable into the class Kfin of all finite members in K. For a variety (equational class) K this is equivalent to the condition that every finitely presented algebra in K is residually finite, i.e., embeddable into a direct product of finite algebras from K ([Evans [1969]).

A class K has the FEP iff if every (G, ·) ∈ K is locally embeddable into the class Kfin of all finite members in K. For a variety (equational class) K this is equivalent to the condition that every finitely presented algebra in K is residually finite, i.e., embeddable into a direct product of finite algebras from K ([Evans [1969]). Groups locally embeddable into (the class of all) finite groups are called LEF groups.

A class K has the FEP iff if every (G, ·) ∈ K is locally embeddable into the class Kfin of all finite members in K. For a variety (equational class) K this is equivalent to the condition that every finitely presented algebra in K is residually finite, i.e., embeddable into a direct product of finite algebras from K ([Evans [1969]). Groups locally embeddable into (the class of all) finite groups are called LEF groups. Unlike the abelian ones, not all groups are LEF, in other words, the class of all groups doesn’t have the FEP.

A class K has the FEP iff if every (G, ·) ∈ K is locally embeddable into the class Kfin of all finite members in K. For a variety (equational class) K this is equivalent to the condition that every finitely presented algebra in K is residually finite, i.e., embeddable into a direct product of finite algebras from K ([Evans [1969]). Groups locally embeddable into (the class of all) finite groups are called LEF groups. Unlike the abelian ones, not all groups are LEF, in other words, the class of all groups doesn’t have the FEP. E.g., the finitely presented Baumslag-Solitar groups BS(m, n) =

• a, b | a−1bma = bn

, for |m|, |n| > 1, are not residually finite, hence not LEF.

A class K has the FEP iff if every (G, ·) ∈ K is locally embeddable into the class Kfin of all finite members in K. For a variety (equational class) K this is equivalent to the condition that every finitely presented algebra in K is residually finite, i.e., embeddable into a direct product of finite algebras from K ([Evans [1969]). Groups locally embeddable into (the class of all) finite groups are called LEF groups. Unlike the abelian ones, not all groups are LEF, in other words, the class of all groups doesn’t have the FEP. E.g., the finitely presented Baumslag-Solitar groups BS(m, n) =

• a, b | a−1bma = bn

, for |m|, |n| > 1, are not residually finite, hence not LEF. A complete list of minimal partial Latin squares embeddable into a closely related infinite group but not embeddable into any finite group, even under a weaker concept of embedding, was recently described by Dietrich and Wanless [2019].

This raises the question of finding some classes of finite grupoids into which all the groups were locally embeddable and which, at the same time, would be “as close to groups as possible”.

This raises the question of finding some classes of finite grupoids into which all the groups were locally embeddable and which, at the same time, would be “as close to groups as possible”. The question is of interest for various reasons: The class of all LEF groups properly extends the class of all locally residually finite groups and plays an important role in dynamical systems, cellular automata, etc.

This raises the question of finding some classes of finite grupoids into which all the groups were locally embeddable and which, at the same time, would be “as close to groups as possible”. The question is of interest for various reasons: The class of all LEF groups properly extends the class of all locally residually finite groups and plays an important role in dynamical systems, cellular automata, etc. Glebsky and Gordon [2005] have shown that a group is locally embeddable into finite semigroups iff it is an LEF group.

This raises the question of finding some classes of finite grupoids into which all the groups were locally embeddable and which, at the same time, would be “as close to groups as possible”. The question is of interest for various reasons: The class of all LEF groups properly extends the class of all locally residually finite groups and plays an important role in dynamical systems, cellular automata, etc. Glebsky and Gordon [2005] have shown that a group is locally embeddable into finite semigroups iff it is an LEF group. It follows that looking for a class of finite groupoids into which

• ne could locally embed all the groups one has to sacrifice the

associativity condition.

This raises the question of finding some classes of finite grupoids into which all the groups were locally embeddable and which, at the same time, would be “as close to groups as possible”. The question is of interest for various reasons: The class of all LEF groups properly extends the class of all locally residually finite groups and plays an important role in dynamical systems, cellular automata, etc. Glebsky and Gordon [2005] have shown that a group is locally embeddable into finite semigroups iff it is an LEF group. It follows that looking for a class of finite groupoids into which

• ne could locally embed all the groups one has to sacrifice the

associativity condition. They also noticed that the results about extendability of partial Latin squares to (complete) Latin squares imply that every group is locally embeddable into finite quasigroups.

This raises the question of finding some classes of finite grupoids into which all the groups were locally embeddable and which, at the same time, would be “as close to groups as possible”. The question is of interest for various reasons: The class of all LEF groups properly extends the class of all locally residually finite groups and plays an important role in dynamical systems, cellular automata, etc. Glebsky and Gordon [2005] have shown that a group is locally embeddable into finite semigroups iff it is an LEF group. It follows that looking for a class of finite groupoids into which

• ne could locally embed all the groups one has to sacrifice the

associativity condition. They also noticed that the results about extendability of partial Latin squares to (complete) Latin squares imply that every group is locally embeddable into finite quasigroups. A slight refinement of their argument shows that every group is locally embeddable into finite loops.

Ziman [2004], building upon the methods of extension of partial Latin squares preserving some symmetry conditions (Cruse [1974], and Lindner [1991]), established the FEP for the class of all loops with antiautomorphic inverses (AAIP loops), i.e., loops with two-sided inverses satisfying the identity (xy)−1 = y−1x−1 .

Ziman [2004], building upon the methods of extension of partial Latin squares preserving some symmetry conditions (Cruse [1974], and Lindner [1991]), established the FEP for the class of all loops with antiautomorphic inverses (AAIP loops), i.e., loops with two-sided inverses satisfying the identity (xy)−1 = y−1x−1 . As a consequence, every group is locally embeddable into finite AAIP loops.

Ziman [2004], building upon the methods of extension of partial Latin squares preserving some symmetry conditions (Cruse [1974], and Lindner [1991]), established the FEP for the class of all loops with antiautomorphic inverses (AAIP loops), i.e., loops with two-sided inverses satisfying the identity (xy)−1 = y−1x−1 . As a consequence, every group is locally embeddable into finite AAIP loops. Quasigroups and loops experts consider the class of all AAIP loops still as a “rather far going extension” of the class of all groups (Dr´ apal).

Ziman [2004], building upon the methods of extension of partial Latin squares preserving some symmetry conditions (Cruse [1974], and Lindner [1991]), established the FEP for the class of all loops with antiautomorphic inverses (AAIP loops), i.e., loops with two-sided inverses satisfying the identity (xy)−1 = y−1x−1 . As a consequence, every group is locally embeddable into finite AAIP loops. Quasigroups and loops experts consider the class of all AAIP loops still as a “rather far going extension” of the class of all groups (Dr´ apal). They find the class of all loops with the inverse property (IP loops), i.e., loops with two-sided inverses satisfying the identities x−1(xy) = y = (yx)x−1 , a much more moderate extension of the class of all groups.

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

• for any x, y ∈ P such that (x, y) ∈ D(P), we have
• x−1, xy
• ,
• xy, y−1

∈ D(P) and x−1(xy) = y, (xy)y−1 = x.

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

• for any x, y ∈ P such that (x, y) ∈ D(P), we have
• x−1, xy
• ,
• xy, y−1

∈ D(P) and x−1(xy) = y, (xy)y−1 = x.

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

• for any x, y ∈ P such that (x, y) ∈ D(P), we have
• x−1, xy
• ,
• xy, y−1

∈ D(P) and x−1(xy) = y, (xy)y−1 = x. We call gaps in P the pairs (a, b) belonging to the set Γ(P) = (P × P) D(P) .

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

• for any x, y ∈ P such that (x, y) ∈ D(P), we have
• x−1, xy
• ,
• xy, y−1

∈ D(P) and x−1(xy) = y, (xy)y−1 = x. We call gaps in P the pairs (a, b) belonging to the set Γ(P) = (P × P) D(P) . A partial IP loop (Q, ∗) is called an extension of a partial IP loop (P, ·) if P ⊆ Q, D(P) ⊆ D(Q), and x · y = x ∗ y whenever (x, y) ∈ D(P).

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

• for any x, y ∈ P such that (x, y) ∈ D(P), we have
• x−1, xy
• ,
• xy, y−1

∈ D(P) and x−1(xy) = y, (xy)y−1 = x. We call gaps in P the pairs (a, b) belonging to the set Γ(P) = (P × P) D(P) . A partial IP loop (Q, ∗) is called an extension of a partial IP loop (P, ·) if P ⊆ Q, D(P) ⊆ D(Q), and x · y = x ∗ y whenever (x, y) ∈ D(P). Alternatively we say, that (P, ·) is embedded in (Q, ∗).

A partial IP loop (P, ·) is a set P endowed with a partial binary operation · defined on a subset D(P) ⊆ P × P, called the domain of the operation ·, satisfying the following three conditions:

• there is an element 1 ∈ P, called the unit of P, such that

(1, x), (x, 1) ∈ D(P) and 1x = x1 = x for all x ∈ P;

• for each x ∈ P there is a unique y ∈ P, called the

inverse of x and denoted by y = x−1, such that (x, y), (y, x) ∈ D(P) and xy = yx = 1;

• for any x, y ∈ P such that (x, y) ∈ D(P), we have
• x−1, xy
• ,
• xy, y−1

∈ D(P) and x−1(xy) = y, (xy)y−1 = x. We call gaps in P the pairs (a, b) belonging to the set Γ(P) = (P × P) D(P) . A partial IP loop (Q, ∗) is called an extension of a partial IP loop (P, ·) if P ⊆ Q, D(P) ⊆ D(Q), and x · y = x ∗ y whenever (x, y) ∈ D(P). Alternatively we say, that (P, ·) is embedded in (Q, ∗). In symbols, (P, ·) ≤ (Q, ∗) or just P ≤ Q.

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L.

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L. Given an IP loop L and a finite set X ⊆ L, we can form the finite partial IP loop P = X ∪ {1} ∪ X−1 by restricting the

• riginal loop operation on L to the set

D(P) = {(x, y) ∈ P × P : xy ∈ P}.

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L. Given an IP loop L and a finite set X ⊆ L, we can form the finite partial IP loop P = X ∪ {1} ∪ X−1 by restricting the

• riginal loop operation on L to the set

D(P) = {(x, y) ∈ P × P : xy ∈ P}. Then P ≤ L. Thus Theorem 1 readily implies

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L. Given an IP loop L and a finite set X ⊆ L, we can form the finite partial IP loop P = X ∪ {1} ∪ X−1 by restricting the

• riginal loop operation on L to the set

D(P) = {(x, y) ∈ P × P : xy ∈ P}. Then P ≤ L. Thus Theorem 1 readily implies Theorem 2. The class of all IP loops has the FEP. Equivalently, every finitely presented IP loop is residually finite.

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L. Given an IP loop L and a finite set X ⊆ L, we can form the finite partial IP loop P = X ∪ {1} ∪ X−1 by restricting the

• riginal loop operation on L to the set

D(P) = {(x, y) ∈ P × P : xy ∈ P}. Then P ≤ L. Thus Theorem 1 readily implies Theorem 2. The class of all IP loops has the FEP. Equivalently, every finitely presented IP loop is residually finite. The second, equivalent formulation of Theorem 2 answers in affirmative the question posed by Evans [1978].

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L. Given an IP loop L and a finite set X ⊆ L, we can form the finite partial IP loop P = X ∪ {1} ∪ X−1 by restricting the

• riginal loop operation on L to the set

D(P) = {(x, y) ∈ P × P : xy ∈ P}. Then P ≤ L. Thus Theorem 1 readily implies Theorem 2. The class of all IP loops has the FEP. Equivalently, every finitely presented IP loop is residually finite. The second, equivalent formulation of Theorem 2 answers in affirmative the question posed by Evans [1978]. As a special case of Theorem 2 we obtain

Theorem 1. Every finite partial IP loop P can be embedded into some finite IP loop L. Given an IP loop L and a finite set X ⊆ L, we can form the finite partial IP loop P = X ∪ {1} ∪ X−1 by restricting the

• riginal loop operation on L to the set

D(P) = {(x, y) ∈ P × P : xy ∈ P}. Then P ≤ L. Thus Theorem 1 readily implies Theorem 2. The class of all IP loops has the FEP. Equivalently, every finitely presented IP loop is residually finite. The second, equivalent formulation of Theorem 2 answers in affirmative the question posed by Evans [1978]. As a special case of Theorem 2 we obtain Theorem 3. Every group can be locally embedded into the class of all finite IP loops. Equivalently, every group can be embedded into some ultraproduct of finite IP loops.

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below.

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P:

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P: O2(P) = {x ∈ P : (x, x) ∈ D(P), x = 1 and xx = 1} ,

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P: O2(P) = {x ∈ P : (x, x) ∈ D(P), x = 1 and xx = 1} , O3(P) = {x ∈ P : (x, x), (x, xx) ∈ D(P), x = 1 and x(xx) = 1} .

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P: O2(P) = {x ∈ P : (x, x) ∈ D(P), x = 1 and xx = 1} , O3(P) = {x ∈ P : (x, x), (x, xx) ∈ D(P), x = 1 and x(xx) = 1} . For x = 1 in P we have x ∈ O2(P) iff x−1 = x,

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P: O2(P) = {x ∈ P : (x, x) ∈ D(P), x = 1 and xx = 1} , O3(P) = {x ∈ P : (x, x), (x, xx) ∈ D(P), x = 1 and x(xx) = 1} . For x = 1 in P we have x ∈ O2(P) iff x−1 = x, and x ∈ O3(P) iff (x, x) ∈ D(P) and x−1 = xx.

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P: O2(P) = {x ∈ P : (x, x) ∈ D(P), x = 1 and xx = 1} , O3(P) = {x ∈ P : (x, x), (x, xx) ∈ D(P), x = 1 and x(xx) = 1} . For x = 1 in P we have x ∈ O2(P) iff x−1 = x, and x ∈ O3(P) iff (x, x) ∈ D(P) and x−1 = xx. Proposition 1. Let P be a finite partial IP loop. Then there exists a finite partial IP loop Q ≥ P such that 3 | #O3(Q).

Thus it suffices to prove just Theorem 1. It is a consequence of the following three propositions to be formulated below. Elements of order 2 and 3, respectively, can be defined for any partial IP loop P: O2(P) = {x ∈ P : (x, x) ∈ D(P), x = 1 and xx = 1} , O3(P) = {x ∈ P : (x, x), (x, xx) ∈ D(P), x = 1 and x(xx) = 1} . For x = 1 in P we have x ∈ O2(P) iff x−1 = x, and x ∈ O3(P) iff (x, x) ∈ D(P) and x−1 = xx. Proposition 1. Let P be a finite partial IP loop. Then there exists a finite partial IP loop Q ≥ P such that 3 | #O3(Q). Proposition 2. Let P be a finite partial IP loop such that 3 | #O3(P). Then there exists a finite partial IP loop Q ≥ P satisfying the following four conditions:

• 3 | #O3(Q), #Q ≥ 10, #Q ≡ 4 (mod 6), and

Γ(Q) ⊆ O2(Q) × O2(Q) .

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P).

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3.

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅.

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q).

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q). If Γ(Q) = ∅ then L = Q is already a finite IP loop extending P.

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q). If Γ(Q) = ∅ then L = Q is already a finite IP loop extending P. Otherwise, by Proposition 2, we obtain a finite partial IP loop Q1 ≥ Q satisfying the four conditions from Proposition 2.

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q). If Γ(Q) = ∅ then L = Q is already a finite IP loop extending P. Otherwise, by Proposition 2, we obtain a finite partial IP loop Q1 ≥ Q satisfying the four conditions from Proposition 2. If Γ(Q1) = ∅ then we are done.

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q). If Γ(Q) = ∅ then L = Q is already a finite IP loop extending P. Otherwise, by Proposition 2, we obtain a finite partial IP loop Q1 ≥ Q satisfying the four conditions from Proposition 2. If Γ(Q1) = ∅ then we are done. Otherwise, we can apply Proposition 3 and get a finite partial IP loop Q2 ≥ Q1 satisfying those four conditions, such that #Γ(Q2) < #Γ(Q1).

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q). If Γ(Q) = ∅ then L = Q is already a finite IP loop extending P. Otherwise, by Proposition 2, we obtain a finite partial IP loop Q1 ≥ Q satisfying the four conditions from Proposition 2. If Γ(Q1) = ∅ then we are done. Otherwise, we can apply Proposition 3 and get a finite partial IP loop Q2 ≥ Q1 satisfying those four conditions, such that #Γ(Q2) < #Γ(Q1). Iterating this step finitely many times we finally arrive at some finite partial IP loop Qn extending P such that Γ(Qn) = ∅.

Proposition 3. Let P be a finite partial IP loop satisfying the last four conditions, such that Γ(P) = ∅. Then there exists a finite partial IP loop Q ≥ P satisfying those four conditions, as well, such that #Γ(Q) < #Γ(P). Theorem 1 follows from Propositions 1, 2 and 3. Let P be a finite partial IP loop P, such that Γ(P) = ∅. Using Proposition 1, we can find a finite partial IP loop Q ≥ P such that 3 | #O3(Q). If Γ(Q) = ∅ then L = Q is already a finite IP loop extending P. Otherwise, by Proposition 2, we obtain a finite partial IP loop Q1 ≥ Q satisfying the four conditions from Proposition 2. If Γ(Q1) = ∅ then we are done. Otherwise, we can apply Proposition 3 and get a finite partial IP loop Q2 ≥ Q1 satisfying those four conditions, such that #Γ(Q2) < #Γ(Q1). Iterating this step finitely many times we finally arrive at some finite partial IP loop Qn extending P such that Γ(Qn) = ∅. Then L = Qn ≥ P is a finite IP loop we have been looking for.

The proofs are technical (almost 12 pages),

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;
• filling in gaps along hamiltonian cycles in graphs with

certain vertex sets V ⊆ P and edges formed by gaps in P, based on Dirac’s criterion for the existence of hamiltonian cycles;

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;
• filling in gaps along hamiltonian cycles in graphs with

certain vertex sets V ⊆ P and edges formed by gaps in P, based on Dirac’s criterion for the existence of hamiltonian cycles;

• Steiner triple systems.

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;
• filling in gaps along hamiltonian cycles in graphs with

certain vertex sets V ⊆ P and edges formed by gaps in P, based on Dirac’s criterion for the existence of hamiltonian cycles;

• Steiner triple systems.

Let P be a partial IP loop, A be a nonempty set such that P ∩ A = ∅, and σ: A → A be an involution, i.e., σ(σ(a)) = a for a ∈ A.

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;
• filling in gaps along hamiltonian cycles in graphs with

certain vertex sets V ⊆ P and edges formed by gaps in P, based on Dirac’s criterion for the existence of hamiltonian cycles;

• Steiner triple systems.

Let P be a partial IP loop, A be a nonempty set such that P ∩ A = ∅, and σ: A → A be an involution, i.e., σ(σ(a)) = a for a ∈ A. The minimal extension P[A, σ] of P by [A, σ] is the partial IP loop with base set P ∪ A,

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;
• filling in gaps along hamiltonian cycles in graphs with

certain vertex sets V ⊆ P and edges formed by gaps in P, based on Dirac’s criterion for the existence of hamiltonian cycles;

• Steiner triple systems.

Let P be a partial IP loop, A be a nonempty set such that P ∩ A = ∅, and σ: A → A be an involution, i.e., σ(σ(a)) = a for a ∈ A. The minimal extension P[A, σ] of P by [A, σ] is the partial IP loop with base set P ∪ A, domain D

• P[A, σ]
• = D(P) ∪
• A × {1}
• ∪
• {1} × A
• ∪
• (a, σ(a)), (σ(a), a) : a ∈ A
• ,

The proofs are technical (almost 12 pages), using mainly

• minimal extensions of partial IP loops;
• filling in gaps along hamiltonian cycles in graphs with

certain vertex sets V ⊆ P and edges formed by gaps in P, based on Dirac’s criterion for the existence of hamiltonian cycles;

• Steiner triple systems.

Let P be a partial IP loop, A be a nonempty set such that P ∩ A = ∅, and σ: A → A be an involution, i.e., σ(σ(a)) = a for a ∈ A. The minimal extension P[A, σ] of P by [A, σ] is the partial IP loop with base set P ∪ A, domain D

• P[A, σ]
• = D(P) ∪
• A × {1}
• ∪
• {1} × A
• ∪
• (a, σ(a)), (σ(a), a) : a ∈ A
• ,

such that 1a = a1 = a and a σ(a) = σ(a) a = 1, i.e., a−1 = σ(a).

The introductory discussion together with Theorem 3 naturally lead to the following question:

The introductory discussion together with Theorem 3 naturally lead to the following question: Problem 1. Is there some minimal (ore even the least) axiomatic class K of IP loops such that every group is locally embeddable into Kfin? Does this class (if it exists) satisfy the Finite Embeddability Property?

The introductory discussion together with Theorem 3 naturally lead to the following question: Problem 1. Is there some minimal (ore even the least) axiomatic class K of IP loops such that every group is locally embeddable into Kfin? Does this class (if it exists) satisfy the Finite Embeddability Property? The first candidate seems to be the class of all Moufang loops, i.e., loops satisfying the identity x(y(xz)) = ((xy)x)z .

The introductory discussion together with Theorem 3 naturally lead to the following question: Problem 1. Is there some minimal (ore even the least) axiomatic class K of IP loops such that every group is locally embeddable into Kfin? Does this class (if it exists) satisfy the Finite Embeddability Property? The first candidate seems to be the class of all Moufang loops, i.e., loops satisfying the identity x(y(xz)) = ((xy)x)z . Every Moufang loop is an IP loop.

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it.

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it. A group (G, ·, 1) is sofic if for every finite set X ⊆ G and every ε > 0 there exists a finite quasigroup (Q, ∗) such that X ⊆ Q, xy = x ∗ y whenever x, y, xy ∈ X, and

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it. A group (G, ·, 1) is sofic if for every finite set X ⊆ G and every ε > 0 there exists a finite quasigroup (Q, ∗) such that X ⊆ Q, xy = x ∗ y whenever x, y, xy ∈ X, and #{q ∈ Q : 1 ∗ q = q} #Q < ε

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it. A group (G, ·, 1) is sofic if for every finite set X ⊆ G and every ε > 0 there exists a finite quasigroup (Q, ∗) such that X ⊆ Q, xy = x ∗ y whenever x, y, xy ∈ X, and #{q ∈ Q : 1 ∗ q = q} #Q < ε as well as #{q ∈ Q : (x ∗ y) ∗ q = x ∗ (y ∗ q)} #Q < ε .

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it. A group (G, ·, 1) is sofic if for every finite set X ⊆ G and every ε > 0 there exists a finite quasigroup (Q, ∗) such that X ⊆ Q, xy = x ∗ y whenever x, y, xy ∈ X, and #{q ∈ Q : 1 ∗ q = q} #Q < ε as well as #{q ∈ Q : (x ∗ y) ∗ q = x ∗ (y ∗ q)} #Q < ε . No example of a non-sofic group is known up today.

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it. A group (G, ·, 1) is sofic if for every finite set X ⊆ G and every ε > 0 there exists a finite quasigroup (Q, ∗) such that X ⊆ Q, xy = x ∗ y whenever x, y, xy ∈ X, and #{q ∈ Q : 1 ∗ q = q} #Q < ε as well as #{q ∈ Q : (x ∗ y) ∗ q = x ∗ (y ∗ q)} #Q < ε . No example of a non-sofic group is known up today. Theorem 3 and the above description indicate that the sofic groups could perhaps be characterized as groups locally embeddable into some “nice” subclass of the class of finite IP loops, fulfilling some “reasonable amount of associativity”.

The following is not the usual definition of the concept of a sofic group (Gromov [1999], Weiss [2000]), however, as proved by Gordon and Glebsky [2007], it is equivalent to it. A group (G, ·, 1) is sofic if for every finite set X ⊆ G and every ε > 0 there exists a finite quasigroup (Q, ∗) such that X ⊆ Q, xy = x ∗ y whenever x, y, xy ∈ X, and #{q ∈ Q : 1 ∗ q = q} #Q < ε as well as #{q ∈ Q : (x ∗ y) ∗ q = x ∗ (y ∗ q)} #Q < ε . No example of a non-sofic group is known up today. Theorem 3 and the above description indicate that the sofic groups could perhaps be characterized as groups locally embeddable into some “nice” subclass of the class of finite IP loops, fulfilling some “reasonable amount of associativity”. A natural candidate is the class of all finite Moufang loops.

One should start with trying to clarify the following question:

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP?

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP? If the answer is negative then it would make sense to elaborate

• n the following problem:

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP? If the answer is negative then it would make sense to elaborate

• n the following problem:

Problem 3. Characterize those groups which are locally embeddable into finite Moufang loops.

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP? If the answer is negative then it would make sense to elaborate

• n the following problem:

Problem 3. Characterize those groups which are locally embeddable into finite Moufang loops. Two possible responses to Problem 3:

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP? If the answer is negative then it would make sense to elaborate

• n the following problem:

Problem 3. Characterize those groups which are locally embeddable into finite Moufang loops. Two possible responses to Problem 3: Conjecture 1. Every group is locally embeddable into finite Moufang loops.

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP? If the answer is negative then it would make sense to elaborate

• n the following problem:

Problem 3. Characterize those groups which are locally embeddable into finite Moufang loops. Two possible responses to Problem 3: Conjecture 1. Every group is locally embeddable into finite Moufang loops. Conjecture 2. A group G is sofic if and only if it is locally embeddable into finite Moufang loops.

One should start with trying to clarify the following question: Problem 2. Does the class of all Moufang loops have the FEP? If the answer is negative then it would make sense to elaborate

• n the following problem:

Problem 3. Characterize those groups which are locally embeddable into finite Moufang loops. Two possible responses to Problem 3: Conjecture 1. Every group is locally embeddable into finite Moufang loops. Conjecture 2. A group G is sofic if and only if it is locally embeddable into finite Moufang loops. Comment: We find the first of them (which would follow from the affirmative answer to Problem 2) more probable to be true than the second one.

Let G be a (hausdorff) locally compact topological group.

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux
• ∀a, b ∈ H : η(a), η(b) ∈ C ⇒ η(a) η(b) ∈ Uη(ab)

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux
• ∀a, b ∈ H : η(a), η(b) ∈ C ⇒ η(a) η(b) ∈ Uη(ab)
• Theorem. (Turing [1938]) A compact connected Lie group G

is approximable by finite groups iff G is abelian.

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux
• ∀a, b ∈ H : η(a), η(b) ∈ C ⇒ η(a) η(b) ∈ Uη(ab)
• Theorem. (Turing [1938]) A compact connected Lie group G

is approximable by finite groups iff G is abelian. A locally compact group G is called unimodular if the left and the right invariant Haar measures on G coincide.

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux
• ∀a, b ∈ H : η(a), η(b) ∈ C ⇒ η(a) η(b) ∈ Uη(ab)
• Theorem. (Turing [1938]) A compact connected Lie group G

is approximable by finite groups iff G is abelian. A locally compact group G is called unimodular if the left and the right invariant Haar measures on G coincide.

• Theorem. (Glebsky, Gordon [2005]) A locally compact group

G is approximable by finite quasigroups iff G is unimodular.

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux
• ∀a, b ∈ H : η(a), η(b) ∈ C ⇒ η(a) η(b) ∈ Uη(ab)
• Theorem. (Turing [1938]) A compact connected Lie group G

is approximable by finite groups iff G is abelian. A locally compact group G is called unimodular if the left and the right invariant Haar measures on G coincide.

• Theorem. (Glebsky, Gordon [2005]) A locally compact group

G is approximable by finite quasigroups iff G is unimodular. “Finite quasigroups” can be replaced by “finite loops.”

Let G be a (hausdorff) locally compact topological group. Then G is approximable by grupoids from a class K if for every compact set C ⊆ G and every neighborhood U of 1 ∈ G there is some H ∈ K and a map η: H → G such that

• ∀ x ∈ C ∃ a ∈ H : η(a) ∈ Ux
• ∀a, b ∈ H : η(a), η(b) ∈ C ⇒ η(a) η(b) ∈ Uη(ab)
• Theorem. (Turing [1938]) A compact connected Lie group G

is approximable by finite groups iff G is abelian. A locally compact group G is called unimodular if the left and the right invariant Haar measures on G coincide.

• Theorem. (Glebsky, Gordon [2005]) A locally compact group

G is approximable by finite quasigroups iff G is unimodular. “Finite quasigroups” can be replaced by “finite loops.” Conjecture 3. Every unimodular locally compact group G is approximable by finite IP loops.

Thank you for your attention.