The government/ society erects a building in memory of or in praise of a person who worked for nation or society. What is that building called?
- a statue or structure established to remind people of a person or event. "a memorial to General Robert E. Lee" synonyms: monument, shrine, mausoleum, cenotaph;
Note the list of synonyms within the definition.
a memorial is something designed to preserve the memory of a person, event, etc., as a monument (or a holiday).
Located in Washington, The Lincoln memorial honors the “Great Emancipator” and 16th President of the United States, Abraham Lincoln. Designed by Henry Bacon in the style of ancient Greek temples, construction began in 1914, with the memorial opening to the public in 1922....
I think you are referring to a monument:
a structure, such as a building or sculpture, erected as a memorial.
something erected in memory of a person, event, etc., as a building, pillar, or statue.
The Free Dictionary
American commemorative monument for the Battle of Saillant de Saint-Mihiel - Montsec.