I read the following article "Finite lattices and Grobner bases" by T. Hibi and J. Herzog. I can't attach the link because it can't be found online.
Many properties from article are proved in
J. Herzog and T. Hibi, Monomial Ideals, GTM Vol. 260 (Springer–Verlag, 2010). [3] T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, in: Commutative Algebra and Combinatorics, edited by M. Nagata and H. Matsumura, Advanced Studies in Pure Math. Vol. 11 (North–Holland, Amsterdam, 1987), pp. 93–109.
Can I find this source online or some elementary sources about distributive lattices, modular lattices, planar lattices, Hibi rings and Grobner bases?
You can find some information about distributive lattices and modular lattices in Stanley's Enumerative Combinatorics, which can be found online.
I found about this article on arxiv https://arxiv.org/pdf/1409.2445.pdf which contains information about Hibi rings and Groebner bases.
If you want to find more about Groebner bases, I recommend the book: D. Cox, J. Little, D. O'Shea - Ideals, Varieties and Algorithms.
Those sources mentioned in the article are very good resources. Too bad you can't find them. Also, it may be useful V. Ene and J. Herzog - Groebner Bases in Commutative Algebra, if you can find it.
Hope these resources are useful!