doc. RNDr. Miroslav Korbelář, Ph.D.

The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space Z(N)). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann-Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that-in N-dimensional quantum mechanics-the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of SL(2, ZN) it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

Let S be a multiplicatively idempotent congruence-simple semiring. We show that |S| = 2 if S has a multiplicatively absorbing element. We also prove that if S is finite then either |S| = 2 or S expressionpproximexpressiontely equexpressionl to End(L) or S-op expressionpproximexpressiontely equexpressionl to End(L) where L is the 2-element semilattice. It seems to be an open question, whether S can be infinite at all.

In this paper, we provide a classification of congruence-simple semirings with a multiplicatively absorbing element and without nontrivial nilpotent elements.

Let P be a finitely generated commutative semiring with a unity. It was shown recently that if the multiplicative reduct of P is a group then P is additively idempotent. We extend this result by showing that P is additively idempotent, provided that P is additively divisible. We further generalize this result using a weaker form of divisibility (almost-divisibility) as follows. Let S be a semiring that is a factor of a monoid semiring N[C] where C is a submonoid of a free commutative monoid of finite rank. Then the semiring S is additively almost-divisible if and only if S is torsion. In particular, we show that if S is a ring then S contains no non-finitely generated subring of Q.

We study additively idempotent congruence-simple semirings with a bi-absorbing element. We characterize a subclass of these semirings in terms of semimodules of a special type (o-characteristic semimodules). We show that o-characteristic semimodules are uniquely determined. We also generalize a result by Jezek and Kepka on simple semirings of endomorphisms of semilattices.

We prove that every ring is a proper homomorphic image of some subdirectly irreducible ring. We also show that a finite ring R does not need to be isomorphic to the factor of a subdirectly irreducible ring by its monolith as well as R does not need to be a homomorphic image of a finite subdirectly irreducible ring. We provide an analogous characterization also for varieties of rings with unity, for the quasiregular rings, for the rings with involution and for their subvarieties of commutative rings.

Let (S,+, .) be a semiring generated by one element. Let us denote this element by w is an element of S and let g(x) is an element of x . N[x] be a polynomial. It has been proved that if g(x) contains at least two different monomials, then the elements of the form g(w) may possibly be contained in any countable commutative semigroup. In particular, divisibility of such elements does not imply their torsion. Let, on the other hand, g(x) consist of a single monomial (i.e. g(x) = kx(n), where k, n is an element of N). We show that in this case, the divisibility of g(w) by infinitely many primes implies that g(w) generates a group within (S, +). Further, an element a is an element of S is called an m-fraction of an element z is an element of S if m is an element of N and z = m . a. We prove that "almost every" m-fraction of w(n) can be expressed as f(w) for some polynomial f is an element of x . N[x] of degree at most n.

We prove that a commutative parasemifield S is additively idempotent, provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties-torsion and almost-divisibility (an element a is an element of S is called almost-divisible in S if there is b is an element of Nsuch that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements f is an element of F(X) such that for every epimorphism pi of F(X) torsion and almost-divisibility of pi(f) are equivalent in pi (F(X)).