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### Systems of Equations Satisfied in All Commutative Finite Semigroups [chapter]

Paweł Parys
Foundations of Software Science and Computational Structures
The following problem is considered: check if a system of equations has a solution in every commutative finite semigroup. It is shown that the problem is decidable, and NP-complete.  ...  However unlike in the commutative case, there are systems of equations satisfied in all finite  ...  A related problem is to check if a system of equations is satisfied in all semigroups simultaneously.  ...

### On the size of independent systems of equations in semigroups

Juhani Karhumäki, Wojciech Plandowski
1996 Theoretical Computer Science
First, in  it was shown that the above compactness property does not hold in all semigroups, in particular it does not hold in the monoid of all finite languages.  ...  Clearly, v E Vi by the definition of <. Using commutativity rules in V, we move all the occurrences of v and I? to the beginning of both sides of equations in E.  ...

### Page 4224 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews
Summary: “We consider systems u; = v; (i € 1) of equations in semigroups over finite sets of variables.  ...  A semigroup (or a monoid) S is said to satisfy the compactness property (CP, for short), if each system of equations has an equivalent finite subsystem.  ...

### The Absolute of finitely generated groups: I. Commutative groups [article]

A.Malyutin, A.Vershik
2018 arXiv   pre-print
We give a complete description of the absolute of commutative finitely generated groups and semigroups.  ...  A central measure (with respect to a finite system of generators of a group or semigroup) is a Markov measure on the space of trajectories whose cotransition distribution at every point is the uniform  ...  Proof of Theorems 4.1-4.3. Let G be a commutative semigroup with a finite system of generators S.  ...

### Page 5669 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews
If the system ® defines a 2; semigroup then & satisfies C (4). Theorem 2. If the system ® defines a 24 semigroup then ® satisfies C(4).  ...  It is formulated in terms of graphs associated to every C, with “points” in C,_; (n € N), where C, is the set of all elements s of the given finitely generated and periodic semigroup S such that n is the  ...

### Page 2060 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews
A survey is given of semigroup conditions which ensure the finite- ness of a finitely generated semigroup S satisfying them. Two results in this direction are as follows: (1) (E.  ...  The authors consider systems of equations AX = BX where the entries from A and B come from NU {0}. They show that if C = A-—B satisfies the columns condition over Z, then the system AX =  ...

### Page 6041 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews
In particular, LI denotes the variety of all finite semigroups such that eSe =e for every idempotent e € S. Let V,,, be the variety of all finite semigroups satisfying the equation x‘ = x‘*?.  ...  If V is a (pseudo)variety of finite semigroups, let LV be the variety of all finite semigroups S such that eSe € V for every idempotent e € S.  ...

### On disjunction of equations in the semigroup language with no constants [article]

Artem N. Shevlyakov
2013 arXiv   pre-print
A semigroup S is an equational domain if any finite union of algebraic sets over S is algebraic. We prove that every nontrivial semigroup in the standard language {·} is not an equational domain.  ...  A system of equations (a system for shortness) is an arbitrary set of equations. The solution set of a system S in the semigroup S is naturally defined and denoted by V S (S).  ...  Introduction It follows from commutative algebra that the union Y of two algebrais sets Y 1 , Y 2 over a field k is algebraic again (i.e. Y is a solution of a system of algebraic equations).  ...

### Page 5131 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews
If A is an S-subsystem of B then A is said to be pure in B if every finite system of equations over A having a solution in B has a solution in A.  ...  If this implication is required to hold only for systems of equations in one variable then A is said to be almost pure in B.  ...

### Page 1408 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews
The complete sequence of equations for J; *Com,, is given, where Com,, is the variety of finite commutative monoids satisfying the equation x!  ...  In this paper we give a description of a free object with a given nonempty basis in a class of commutative (n,m)-semigroups defined by a system of identities of the form [x/] = [v7], where p,q > m and  ...

### Page 179 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews
While finitely generated soluble groups with solvable word prob- lem can have undecidable equational theory, the equational theory for finitely generated soluble groups satisfying a semigroup identity  ...  Suppose the semigroup identity u = v is satisfied by the semigroup S which itself is a generating set in a group G. Is then wu =v an identity satisfied by the group G?  ...

### Page 927 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews
In addition, the presentation is applied in solving systems of equations on finitely generated commutative Celia L. Adair (Spartanburg, SC) monoids.  ...  All the semigroups appearing in this paper are commutative and cancellative (this last kind of semigroups are also known as D- semigroups).  ...

### Page 2643 of Mathematical Reviews Vol. , Issue 90E [page]

1990 Mathematical Reviews
An S-variety is a collection of finite semigroups closed under finite direct products and division. The S-variety D is the collection of all finite semigroups S that satisfy Se = e for each e?  ...  Let C, = AoGoA; A is the M-variety of all finite aperiodic monoids (every group divisor is trivial) and G is the M-variety of all finite groups. Rhodes showed that LC, # C, oD.  ...

### On commutative, nonpotent archimedean semigroups

Richard G. Levin
1968 Pacific Journal of Mathematics
The following is a crucial lemma in the proof of the main theorem: let S be a finitely generated, commutative, nonpotent, archimedean semigroup; then the set of maximal elements of S relative to ^5 is  ...  Proofs of all other results in this paper are supplied.  ...

### Lectures notes in universal algebraic geometry [article]

A. Shevlyakov
2016 arXiv   pre-print
Lectures notes in universal algebraic geometry for beginners  ...  Surprisingly, the solution sets of linear equations in free anti-commutative algebra is simpler than the corresponding sets over free Lie algebra (see  ).  ...  Thus, it is impossible to obtain a nice description of all coordinate algebras of algebraic sets over free Lie algebras.  ...
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