[Grätzer and Schmidt 1963] proves that every algebraic lattice is isomorphic to the congruence lattice of a universal algebra. A finite lattice is algebraic. The finite lattice representation problem asks whether every finite lattice is the congruence lattice of a finite universal algebra. The answer is generally suspected to be no.
Question 1: What is the smallest lattice which is not known to be a congruence lattice of a finite universal algebra?
Remark: By “smallest” I mean “of smallest cardinal”. By “not known” I don’t mean that it should have been conjectured somewhere to be a counter-example.
[Pálfy and Pudlák 1980] proves that the above problem is equivalent to ask whether every finite lattice is the lattice of an interval in the subgroup lattice of a finite group. As pointed out by the authors, this theorem does not imply that every congruence lattice of a finite universal algebra is the lattice of an interval in the subgroup lattice of a finite group (whereas the converse is true), it is just an equivalence between two problems.
Question 2: What is the smallest lattice which is not known to be the lattice of an interval in the subgroup lattice of a finite group?
Remark: same as above.
This short course by William DeMeo provides a clear review of the background for this problem.
✓ Extra quality
ExtraProxies brings the best proxy quality for you with our private and reliable proxies
✓ Extra anonymity
Top level of anonymity and 100% safe proxies – this is what you get with every proxy package
✓ Extra speed
1,ooo mb/s proxy servers speed – we are way better than others – just enjoy our proxies!
USA proxy location
We offer premium quality USA private proxies – the most essential proxies you can ever want from USA
Our proxies have TOP level of anonymity + Elite quality, so you are always safe and secure with your proxies
Use your proxies as much as you want – we have no limits for data transfer and bandwidth, unlimited usage!
Superb fast proxy servers with 1,000 mb/s speed – sit back and enjoy your lightning fast private proxies!
99,9% servers uptime
Alive and working proxies all the time – we are taking care of our servers so you can use them without any problems
No usage restrictions
You have freedom to use your proxies with every software, browser or website you want without restrictions
Perfect for SEO
We are 100% friendly with all SEO tasks as well as internet marketing – feel the power with our proxies
Buy more proxies and get better price – we offer various proxy packages with great deals and discounts
We are working 24/7 to bring the best proxy experience for you – we are glad to help and assist you!